A Parallel Adaptive Cartesian PDE Solver Using Space–Filling Curves

Bungartz, Hans‐Joachim; Mehl, Miriam; Weinzierl, Tobias

doi:10.1007/11823285_112

Cited by 38 publications

(33 citation statements)

References 11 publications

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“…To optimize cache-efficiency of the code, Peano uses streams and stacks as the only types of data structures [14,18]. This data concept relies on the palindrome and projection properties of the Peano curve, properties that we could not show for any Hilbert curve in 3D.…”

Section: The Peano Gridmentioning

confidence: 99%

Navier–Stokes and Lattice–Boltzmann on octree‐like grids in the Peano framework

Mehl

Neckel

Neumann

2010

Numerical Methods in Fluids

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SUMMARYThe Navier-Stokes equations (NS) and the Lattice-Boltzmann method (LBM) are the main two types of models used in computational fluid dynamics yielding similar results for certain classes of flow problems both having certain advantages and disadvantages.We present the realization of both approaches-laminar incompressible NS and LB-on the same code basis, our PDE framework Peano. Peano offers a highly memory efficient implementation of all grid and data handling issues for structured adaptive grids. Using a common code basis allows to compare NS and LB without distorting the results by differences in the maturity and degree of hardware-optimality of the technical implementation.In addition to a comparison for some test examples, we briefly present the coupling of NS and LB in one single simulation. Such a coupling might be useful to simulate boundary effects happening on a smaller scale more accurately with LB but still using NS in the main part of the domain.

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Section: The Peano Gridmentioning

confidence: 99%

Navier–Stokes and Lattice–Boltzmann on octree‐like grids in the Peano framework

Mehl

Neckel

Neumann

2010

Numerical Methods in Fluids

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“…Figure 5 shows a twodimensional adaptive quadtree mesh ordered according to the Z-curve together with the cell tree and the linearised representation. In scientific simulation software, the combination of tree-structured grids and space-filling curves has been used in several ways, for example augmented by hashing [41], or for partial differential equation solvers with cache-optimised data administration [42,5]. Octor [43] and Dendro [6] are two examples of parallel octree libraries that have been scaled to 62,000 [44] and 32,000 [45] cores, operating on parent-child pointers and a linearised octant storage, respectively.…”

Section: Parallel Tree-structured Gridsmentioning

confidence: 99%

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

Lahnert¹,

Burstedde²,

Holm³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We present the minimally-invasive exchange of the regular Cartesian grid in the lattice-Boltzmann solver of ESPResSo by a dynamically-adaptive octree grid. Octree grids are favoured by computer scientists over other grid types as they are very memory-efficient. In addition, they represent a natural generalisation of regular Cartesian grids, such that most discretisation details of a regular grid solver can be maintained. Optimised codes, however, require a special tree-oriented grid traversal, which typically conflicts with existing simulation codes using various iterators, some for only parts of the grid, e.g., boundaries. ESPResSo is a large software package developed for soft-matter simulations involving fluid flow, electrostatic, and electrokinetic effects, and molecular dynamics. The currently used regular Cartesian grid hinders the simulation of realistic domain sizes and significant time periods, a problem that can be solved using grid adaptivity. In a first step, we focus on the lattice-Boltzmann flow solver in ESPResSo.p4est is a grid framework, that already provides dynamically adaptive quadtree and octree grids together with high-level interfaces for flexible grid traversals with direct neighbour access in all grid components. In this paper, we first describe extensions of p4est that were necessary to fulfill certain application requirements. The second part of our work consists of the minimally-invasive changes in ESPResSo preserving the expertise accumulated in the software's implementation over years. Our numerical results demonstrate physical correctness of the implementation, good parallel scalability and low overhead of the dynamical grid adaptivity. These are prerequisites to actually profit from grid adaptivity in terms of being able to simulate larger domains over longer time periods with limited computational resources. Thus, the current status forms a solid basis for further steps such as the development of refinement criteria, the setup of more realistic application scenarios, and a GPU implementation.

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“…Today, this principle has found its way into widely used software libraries [5,6]. Memory usage can be further optimized by incremental encoding along the SFC [4,13,55].…”

Section: Introductionmentioning

confidence: 99%

A Tetrahedral Space-Filling Curve for Nonconforming Adaptive Meshes

Burstedde¹,

Holke²

2016

SIAM J. Sci. Comput.

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Abstract. We introduce a space-filling curve for triangular and tetrahedral red-refinement that can be computed using bitwise interleaving operations similar to the well-known Z-order or Morton curve for cubical meshes. To store sufficient information for random access, we define a low-memory encoding using 10 bytes per triangle and 14 bytes per tetrahedron. We present algorithms that compute the parent, children, and face-neighbors of a mesh element in constant time, as well as the next and previous element in the space-filling curve and whether a given element is on the boundary of the root simplex or not. Our presentation concludes with a scalability demonstration that creates and adapts selected meshes on a large distributed-memory system. Key words. Forest of octrees, parallel adaptive mesh refinement, Morton code, high performance computing, nonconforming simplicial mesh, space-filling curve AMS subject classifications. 65M50, 68W10, 65Y05, 65D181. Introduction. Conforming adaptive mesh refinement for simplicial (triangular and tetrahedral) meshes is one of the most successful concepts in numerical mathematics and computational science and engineering; see, e.g., [3,21,50]. Simplices provide high flexibility in meshing to arbitrary domain geometries [46,47] and can be mapped to a reference simplex using an elementwise constant Jacobian in most cases, which allows for an efficient numerical implementation. Discretization and integration methods of various kinds are available for both low and high orders of accuracy.Large-scale scientific computing requires fast and scalable algorithms for (1) adaptive refinement and coarsening (AMR) as well as (2) parallel partitioning. One class of methods for AMR exploits the properties of Delaunay triangulations [23,26,45], while partitioning of unstructured meshes is often approached by formulating the mesh topology as a graph. These triangulations are usually conforming; that is, elements intersect only along whole faces and edges. [20]. Graph-based algorithms have been advanced to target millions of processes and billions of elements [18,42,49]. Still, increasing the scalability and decreasing the absolute runtime and memory demands of distributed implementations remains a challenge, and the lack of an obvious parent-child structure in many unstructured meshing approaches prevents certain use cases.Nonconforming AMR in combination with recursive refinement makes refinement and coarsening nearly trivial operations. The additional mathematical logic to enable hanging faces and edges is well understood for both continuous and discontinuous discretizations [1,22,30,43]. It is local to the loop over the finite elements or volumes and transparent to most of the numerical pipeline, thus offering the possibility to extend existing conforming codes. The resolution may be as coarse as any chosen root mesh (a mesh which is not intended for further coarsening), which poses only a slight limit to geometric flexibility.The challenge of efficient partitioning of meshes may be addre...

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A Parallel Adaptive Cartesian PDE Solver Using Space–Filling Curves

Cited by 38 publications

References 11 publications

Navier–Stokes and Lattice–Boltzmann on octree‐like grids in the Peano framework

Navier–Stokes and Lattice–Boltzmann on octree‐like grids in the Peano framework

Towards Lattice-Boltzmann on Dynamically Adaptive Grids — Minimally-Invasive Grid Exchange in Espresso

A Tetrahedral Space-Filling Curve for Nonconforming Adaptive Meshes

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