Simulation of incompressible viscous flows on distributed Octree grids

This dissertation focuses on numerical simulation methods for continuous problems with irregular interfaces. A common feature of these types of systems is the locality of the physical phenomena, suggesting the use of adaptive meshes to better focus the computational effort, and the complexity inherent to representing a moving irregular interface. We address these challenges by using the implicit framework provided by the Level-Set method and implemented on adaptive Quadtree (in two spatial dimensions) and Octree (in three spatial dimensions) grids. This work is composed of two sections.In the first half, we present the numerical tools for the study of incompressible monophasic viscous flows. After a study of an alternative grid storage structure to the Quad/Oc-tree data structure based on hash tables, we introduce the extension of the level-set method to massively parallel forests of Octrees. We then detail the numerical scheme developed to attain second order accuracy on non-graded Quad/Oc-tree grids and demonstrate the validity and robustness of the resulting solver. Finally, we combine the fluid solver and the parallel framework together and illustrate the potential of the approach.The second half of this dissertation presents the Voronoi Interface Method (VIM), a new method for solving elliptic systems with discontinuities on irregular interfaces such as the ones encountered when simulating viscous multiphase flows. The VIM relies on a Voronoi mesh built on an underlying Cartesian grid and is compact and second order accurate while preserving the symmetry and positiveness of the resulting linear system. We then compare the VIM with the popular Ghost Fluid Method before adapting it to the simulation of the problem of the electropermeabilization of cells

A cell-centred finite volume method for the Poisson problem on non-graded quadtrees with second order accurate gradients

The final publication is available at Elsevier via https://doi.org/10.1016/j.jcp.2016.11.035 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper introduces a two-dimensional cell-centred finite volume discretization of the Poisson problem on adaptive Cartesian quadtree grids which exhibits second order accuracy in both "the solution and its gradients, and requires no grading condition between adjacent cells. At T-junction configurations, which occur wherever resolution differs between neighboring cells, use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation. To properly recover second order accuracy in the resulting numerical gradients, prior work addressing block-structured grids and graded trees has shown that quadratic, rather than linear, interpolation is required; the gradients otherwise exhibit only first order convergence, which limits potential applications such as fluid flow. However, previous schemes fail or lose accuracy in the presence of the more complex T-junction geometries arising in the case of general non-graded quadtrees, which place no restrictions on the resolution of neighboring cells. We therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed. The method handles complex tree topologies and large resolution jumps between neighboring cells, even along the domain boundary, and both Dirichlet and Neumann boundary conditions are supported. Numerical experiments confirm the overall second order accuracy of the method in the L-infinity norm. (C) 2016 Elsevier Inc. All rights reserved.This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada (Grant RGPIN-04360-2014)

Power Diagrams and Sparse Paged Grids for High Resolution Adaptive Liquids

© ACM, 2017. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Aanjaneya, M., Gao, M., Liu, H., Batty, C., & Sifakis, E. (2017). Power Diagrams and Sparse Paged Grids for High Resolution Adaptive Liquids. ACM Trans. Graph., 36(4), 140:1–140:12. https://doi.org/10.1145/3072959.3073625We 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 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 enhancements to adaptive level set advection, velocity extrapolation, and the fast marching method for redistancing. Our overall framework gracefully accommodates the task of dynamically adapting the octree topology during simulation. We demonstrate end-to-end simulations of complex adaptive flows in irregularly shaped domains, with tens of millions of degrees of freedom.National Science FoundationNational Sciences and Engineering Research Council of Canad

Projection methods for incompressible flow problems with WENO finite difference schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary con\-vec\-tion-diffusion equations [20]. This paper explores the applicability of these sche\-mes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious

An adaptive variational finite difference framework for efficient symmetric octree viscosity

While pressure forces are often the bottleneck in (near-)inviscid fluid simulations, viscosity can impose orders of magnitude greater computational costs at lower Reynolds numbers. We propose an implicit octree finite difference discretization that significantly accelerates the solution of the free surface viscosity equations using adaptive staggered grids, while supporting viscous buckling and rotation effects, variable viscosity, and interaction with scripted moving solids. In experimental comparisons against regular grids, our method reduced the number of active velocity degrees of freedom by as much as a factor of 7.7 and reduced linear system solution times by factors between 3.8 and 9.4. We achieve this by developing a novel adaptive variational finite difference methodology for octrees and applying it to the optimization form of the viscosity problem. This yields a linear system that is symmetric positive definite by construction, unlike naive finite difference/volume methods, and much sparser than a hypothetical finite element alternative. Grid refinement studies show spatial convergence at first order in L∞ and second order in L1, while the significantly smaller size of the octree linear systems allows for the solution of viscous forces at higher effective resolutions than with regular grids. We demonstrate the practical benefits of our adaptive scheme by replacing the regular grid viscosity step of a commercial liquid simulator (Houdini) to yield large speed-ups, and by incorporating it into an existing inviscid octree simulator to add support for viscous flows. Animations of viscous liquids pouring, bending, stirring, buckling, and melting illustrate that our octree method offers significant computational gains and excellent visual consistency with its regular grid counterpart.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (RGPIN-04360-2014, CRDPJ-499952-2016) and the Rutgers University start-up grant

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices. A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces. The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian–Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases

High order time integration and mesh adaptation with error control for incompressible Navier-Stokes and scalar transport resolution on dual grids

International audienceRelying on a building block developed by the authors in order to resolve the incompressible Navier-Stokes equation with high order implicit time stepping and dynamic mesh adaptation based on multiresolution analysis with collocated variables, the present contribution investigates the ability to extend such a strategy for scalar transport at relatively large Schmidt numbers using a finer level of refinement compared to the resolution of the hydrody-namic variables, while preserving space adaptation with error control. This building block is a key part of a strategy to construct a low-Mach number code based on a splitting strategy for combustion applications, where several spatial scales are into play. The computational efficiency and accuracy of the proposed strategy is assessed on a well-chosen three-vortex simulation

Numerical Simulations of the Two-phase flow and Fluid-Structure Interaction Problems with Adaptive Mesh Refinement

Numerical simulations of two-phase flow and fluid structure interaction problems are of great interest in many environmental problems and engineering applications. To capture the complex physical processes involved in these problems, a high grid resolution is usually needed. However, one does not need or maybe cannot afford a fine grid of uniformly high resolution across the whole domain. The need to resolve local fine features can be addressed by the adaptive mesh refinement (AMR) method, which increases the grid resolution in regions of interest as needed during the simulation while leaving general estimates in other regions. In this work, we propose a block-structured adaptive mesh refinement (BSAMR) framework to simulate two-phase flows using the level set (LS) function with both the subcycling and non-subcycling methods on a collocated grid. To the best of our knowledge, this is the first framework that unifies the subcycling and non-subcycling methods to simulate two-phase flows. The use of the collocated grid is also the first among the two-phase BSAMR framework, which significantly simplifies the implementation of multi-level differential operators and interpolation schemes. We design the synchronization operations, including the averaging, refluxing, and synchronization projection, which ensures that the flow field is divergence-free on the multi-level grid. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation.Comment: 178 page

A practical octree liquid simulator with adaptive surface resolution

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 the author(s) 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 [email protected]. © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. 0730-0301/2020/7-ART32 $15.00 https://doi.org/10.1145/3386569.3392460We propose a new adaptive liquid simulation framework that achieves highly detailed behavior with reduced implementation complexity. Prior work has shown that spatially adaptive grids are efficient for simulating large-scale liquid scenarios, but in order to enable adaptivity along the liquid surface these methods require either expensive boundary-conforming (re-)meshing or elaborate treatments for second order accurate interface conditions. This complexity greatly increases the difficulty of implementation and maintainability, potentially making it infeasible for practitioners. We therefore present new algorithms for adaptive simulation that are comparatively easy to implement yet efficiently yield high quality results. First, we develop a novel staggered octree Poisson discretization for free surfaces that is second order in pressure and gives smooth surface motions even across octree T-junctions, without a power/Voronoi diagram construction. We augment this discretization with an adaptivity-compatible surface tension force that likewise supports T-junctions. Second, we propose a moving least squares strategy for level set and velocity interpolation that requires minimal knowledge of the local tree structure while blending near-seamlessly with standard trilinear interpolation in uniform regions. Finally, to maximally exploit the flexibility of our new surface-adaptive solver, we propose several novel extensions to sizing function design that enhance its effectiveness and flexibility. We perform a range of rigorous numerical experiments to evaluate the reliability and limitations of our method, as well as demonstrating it on several complex high-resolution liquid animation scenarios.This research was supported by the JSPS Grant-in-Aid for Young Scientists (18K18060) and the Natural Sciences and Engineering Research Council of Canada (Grant RGPIN-04360-2014)