Simulação de fluidos com PIC usando RBF-FD e grades adaptativas balanceadas

Abstract

In physics based animations, hybrid fluid simulations --- simulations that rely on both material and spatial discretization --- are usually based on a regular subdivision of the domain by using a regular Cartesian grid. Depending on the adopted resolution, such an approach can create a large number of cells that do not contain fluid but will still be processed. To avoid this issue, adaptive grids have been introduced to reduce the number of non-fluid cells. With a Cartesian regular grid, the differential operators needed for a numerical fluid solver can be computed using the finite difference method (FDM). FDM gives an approximation of the partial derivatives of a quantity at a grid point based on the quantities of its neighboring grid points. The configuration formed by the target point and its neighbors is called a stencil. However, a stencil formed by the points of an adaptive grid cannot use FDM for computing the differential operators since such points do not form vectors aligned to the domain Cartesian grid. In this case, an alternative is to use the finite difference based on radial basis functions, which enable stencils in generic configurations. To maintain the benefits of using adaptive grids and accelerate the computing of the differential operator with RBF-FD, this thesis proposes a PIC-based fluid solver that employs a graded adaptive grid, that is, one in which the level difference between two neighboring cells is not greater than one. The proposed approach relies on maps whose entries store the RBF weights for a given differential operator and stencil. The map entries are indexed by a key that encodes the cell level and the relative position of the stencil target and its neighbors. The weights are computed only once and reused for the same stencils throughout the simulation.