Power diagrams and sparse paged grids for high resolution adaptive liquids

Abstract

Fig. 1. (Left)Water filling a river bed surrounded by a canyon, with effective resolution 512 2 × 1024. Three refinement levels are used, based on proximity to the terrain. (Right) Sources inject water into a container and collide to form a thin sheet, with effective resolution 512 3 . Adaptivity pattern shown on background.We present an efficient and scalable octree-inspired fluid simulation framework with the flexibility to leverage adaptivity in any part of the computational domain, even when resolution transitions reach the free surface. Our methodology ensures symmetry, definiteness and second order accuracy of the discrete Poisson operator, and eliminates numerical and visual artifacts of prior octree schemes. This is achieved by adapting the operators acting on the octree's simulation variables to reflect the structure and connectivity of a power diagram, which recovers primal-dual mesh orthogonality and eliminates problematic T-junction configurations. We show how such operators can be efficiently implemented using a pyramid of sparsely populated uniform grids, enhancing the regularity of operations and facilitating parallelization. A novel scheme is proposed for encoding the topology of the power diagram in the neighborhood of each octree cell, allowing us to locally reconstruct it on the fly via a lookup table, rather than resorting to costly explicit meshing. The pressure Poisson equation is solved via a highly efficient, matrix-free multigrid preconditioner for Conjugate Gradient, adapted to the power diagram discretization. We use another sparsely † M. Aanjaneya and M. Gao are joint first authors. * M. Aanjaneya was with the University of Wisconsin -Madison during this work. This work was supported in part by National Science Foundation grants IIS-1253598, CCF-1423064, CCF-1533885 and by the Natural Sciences and Engineering Research Council of Canada under grant RGPIN-04360-2014. The authors are grateful to Nathan Mitchell for his indispensable help with modeling and rendering of examples. C. Batty would like to thank Ted Ying for carrying out preliminary explorations on quadtrees. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. populated uniform grid for high resolution interface tracking with a narrow band level set representation. Using the recently introduced SPGrid data structure, sparse uniform grids in both the power diagram discretization and our narrow band level set can be compactly stored and efficiently updated via streaming operations. Additionally, we present enhancem...

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