PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures

We present a PDE-based approach for the multidimensional extrapolation of smooth scalar quantities across interfaces with kinks and regions of high curvature. Unlike the commonly used method of [2] in which normal derivatives are extrapolated, the proposed approach is based on the extrapolation and weighting of Cartesian derivatives. As a result, second- and third-order accurate extensions in the $L^\infty$ norm are obtained with linear and quadratic extrapolations, respectively, even in the presence of sharp geometric features. The accuracy of the method is demonstrated on a number of examples in two and three spatial dimensions and compared to the approach of [2]. The importance of accurate extrapolation near sharp geometric features is highlighted on an example of solving the diffusion equation on evolving domains.Comment: 17 pages, 13 figures, submitted to SIAM Journal of Scientific Computin

Imposing mixed Dirichlet–Neumann–Robin boundary conditions in a level-set framework

Pre-print (óritrýnt handrit)We consider the Poisson equation with mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains. We describe a straightforward and efficient approach for imposing the mixed boundary conditions using a hybrid finite-volume/finite-difference approach, leveraging on the work of Gibou et al. (2002) [14], Ng et al. (2009) [30] and Papac et al. (2010) [33]. We utilize three different level set functions to represent the irregular boundary at which each of the three different boundary conditions must be imposed; as a consequence, this approach can be applied to moving boundaries. The method is straightforward to implement, produces a symmetric positive definite linear system and second-order accurate solutions in the L-infinity-norm in two and three spatial dimensions. Numerical examples illustrate the second-order accuracy and the robustness of the method. (C) 2015 Elsevier Ltd. All rights reserved.The research of Á. Helgadóttir, Y.T. Ng and F. Gibou were supported in part by ONR under grant agreement N00014-11-1-0027, by the National Science Foundation under grant agreement CHE 1027817 and by the W.M. Keck Foundation. The research of C. Min was supported in part by the Kyung Hee University Research Fund (KHU-20070608) in 2007 and by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-C00045)

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)

A Parallel Adaptive Mesh Refinement Library for Cartesian Meshes

abstract: This dissertation introduces FARCOM (Fortran Adaptive Refiner for Cartesian Orthogonal Meshes), a new general library for adaptive mesh refinement (AMR) based on an unstructured hexahedral mesh framework. As a result of the underlying unstructured formulation, the refinement and coarsening operators of the library operate on a single-cell basis and perform in-situ replacement of old mesh elements. This approach allows for h-refinement without the memory and computational expense of calculating masked coarse grid cells, as is done in traditional patch-based AMR approaches, and enables unstructured flow solvers to have access to the automated domain generation capabilities usually only found in tree AMR formulations. The library is written to let the user determine where to refine and coarsen through custom refinement selector functions for static mesh generation and dynamic mesh refinement, and can handle smooth fields (such as level sets) or localized markers (e.g. density gradients). The library was parallelized with the use of the Zoltan graph-partitioning library, which provides interfaces to both a graph partitioner (PT-Scotch) and a partitioner based on Hilbert space-filling curves. The partitioned adjacency graph, mesh data, and solution variable data is then packed and distributed across all MPI ranks in the simulation, which then regenerate the mesh, generate domain decomposition ghost cells, and create communication caches. Scalability runs were performed using a Leveque wave propagation scheme for solving the Euler equations. The results of simulations on up to 1536 cores indicate that the parallel performance is highly dependent on the graph partitioner being used, and differences between the partitioners were analyzed. FARCOM is found to have better performance if each MPI rank has more than 60,000 cells.Dissertation/ThesisDoctoral Dissertation Aerospace Engineering 201