An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

Olshanskii, Maxim A.; Terekhov, Kirill M.; Vassilevski, Yuri

doi:10.1016/j.compfluid.2013.04.027

Cited by 38 publications

(38 citation statements)

References 38 publications

(70 reference statements)

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“…Additionally, we have focused on the Linux programming interface for our current implementation; as future work we will produce a port of the SPGrid library that also runs under the Windows operating system. Finally, our smoke simulation scheme was very elementary, and we did not leverage opportunities for more accurate advection schemes, methods to counteract numerical dissipation [Olshanskii et al 2013], and higher order accurate discretizations of the Poisson operator [Losasso et al 2005]. We will address such simulation features, along with an investigation of applying SPGrid to free surface flows in future work.…”

Section: Limitations and Future Workmentioning

confidence: 98%

SPGrid

et al. 2014

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Figure 1: Smoke flow past sphere with 135M active voxels, 1K×1K×2K maximum resolution. Adaptive grid shown on the right. AbstractWe introduce a new method for fluid simulation on high-resolution adaptive grids which rivals the throughput and parallelism potential of methods based on uniform grids. Our enabling contribution is SPGrid, a new data structure for compact storage and efficient stream processing of sparsely populated uniform Cartesian grids. SPGrid leverages the extensive hardware acceleration mechanisms inherent in the x86 Virtual Memory Management system to deliver sequential and stencil access bandwidth comparable to dense uniform grids. Second, we eschew tree-based adaptive data structures in favor of storing simulation variables in a pyramid of sparsely populated uniform grids, thus avoiding the cost of indirect memory access associated with pointer-based representations. We show how the costliest algorithmic kernels of fluid simulation can be implemented as a composition of two kernel types: (a) stencil operations on a single sparse uniform grid, and (b) structured data transfers between adjacent levels of resolution, even when modeling non-graded octrees. Finally, we demonstrate an adaptive multigridpreconditioned Conjugate Gradient solver that achieves resolutionindependent convergence rates while admitting a lightweight implementation with a modest memory footprint. Our method is complemented by a new interpolation scheme that reduces dissipative effects and simplifies dynamic grid adaptation. We demonstrate the efficacy of our method in end-to-end simulations of smoke flow.

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Section: Limitations and Future Workmentioning

confidence: 98%

SPGrid

et al. 2014

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“…fu.t C t/; p.t C t/g. This is done in the following substeps: (i) The convective terms are approximated using the second order accurate compact upwind method on octree grids from [12] and the momentum equation is solved for an intermediate velocity, while the plasticity terms are treated explicitly [11]; (ii) The intermediate velocity field is projected onto the subspace of discrete divergence free functions and the pressure is updated by the solution of the discrete Poisson equation.…”

Section: Methodsmentioning

confidence: 99%

“…However, building accurate and stable discretizations on such grids is a challenging task. For the fluid equations we use the second order accurate finite difference/finite volume method with compact nodal stencils developed in [12]. This method is a stable extension of the classical MAC scheme on octree meshes.…”

Section: Introductionmentioning

confidence: 99%

“…This scheme decouples one time step on convection, diffusion and plasticity, pressure correction, and the level set function advection substeps. Numerical stability and accuracy of the whole method were verified for the case of Newtonian flows in [10,12] by computing analytical solutions and benchmark flows in a cubic cavity and over a 3D cylinder. Computed results for a collapsing water column perfectly match available experimental data.…”

Section: Introductionmentioning

confidence: 99%

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A Unified Approach for Computing Tsunami, Waves, Floods, and Landslides

Danilov

Nikitin

Olshanskii

et al. 2014

Lecture Notes in Computational Science and Engineering

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The prediction of large-scale hydrodynamic events such as tsunami spread and run-up, dam break, flood, or landslide run-out is a challenging and important problem of applied mathematics and scientific computing. The paper presents a computational approach based on free surface flow models for fluids of complex rheology to simulate such events and phenomena with detail and prediction confidence typically not achievable by simplified models. Using nonlinear defining relations for stress and rate of strain tensors allows a unified approach to simulate events described by both the Newtonian model (tsunami, dam break) and non-Newtonian models (landslide, snow avalanches, lava flood, mud flow). The computational efficiency of the numerical approach owes to the level-set method for free surface capturing and to an accurate and stable FV/FD method on dynamically adapted octree meshes for discretization of flow and level set equations. In this paper we briefly describe the numerical method and present results of several simulations of hydrodynamic events: a dam break, a landslide and tsunami spread and run-up.

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“…When the free surface evolves, the grid is dynamically re ned or coarsened according to the distance to the free boundary. We note that discretizations on octree (quadtree) cartesian grids already enjoyed an employment in free surface ow computations [24,27,28,32,38], yet many aspects of using octree grids for accurate and predictive computations of free surface ows require more thorough study. To discretize divergence, gradient and Laplace operators, we use nite di erence approximations with compact stencils.…”

Section: Introductionmentioning

confidence: 99%

A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

Nikitin

Olshanskii

Terekhov

et al. 2014

Computational Methods in Applied Mathematics

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The paper studies a splitting method for the numerical time-integration of the system of partial di erential equations describing the motion of viscous incompressible uid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and uid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. E cient computations of free surface ows of incompressible viscous uids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

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An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

Cited by 38 publications

References 38 publications

SPGrid

SPGrid

A Unified Approach for Computing Tsunami, Waves, Floods, and Landslides

A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

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