Extreme-Scale AMR

Burstedde, Carsten; Ghattas, Omar; Gurnis, Michael; Isaac, Tobin; Stadler, Georg; Warburton, Tim; Wilcox, Lucas C.

doi:10.1109/sc.2010.25

Cited by 76 publications

(61 citation statements)

References 35 publications

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“…This behavior is most marked in the Copy-todeal.II and p4est re-partitioning operations. (The scalability of the latter has been independently demonstrated in Burstedde et al [2010]. )…”

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confidence: 95%

Algorithms and data structures for massively parallel generic adaptive finite element codes

Bangerth

Burstedde

Heister

et al. 2011

ACM Trans. Math. Softw.

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175

154

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Today's largest supercomputers have 100,000s of processor cores and offer the potential to solve partial differential equations discretized by billions of unknowns. However, the complexity of scaling to such large machines and problem sizes has so far prevented the emergence of generic software libraries that support such computations, although these would lower the threshold of entry and enable many more applications to benefit from large-scale computing.We are concerned with providing this functionality for mesh-adaptive finite element computations. We assume the existence of an "oracle" that implements the generation and modification of an adaptive mesh distributed across many processors, and that responds to queries about its structure. Based on querying the oracle, we develop scalable algorithms and data structures for generic finite element methods. Specifically, we consider the parallel distribution of mesh data, global enumeration of degrees of freedom, constraints, and postprocessing. Our algorithms remove the bottlenecks that typically limit large-scale adaptive finite element analyses.We demonstrate scalability of complete finite element workflows on up to 16,384 processors. An implementation of the proposed algorithms, based on the open source software p4est as mesh oracle, is provided under an open source license through the widely used deal.II finite element software library.

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“…This behavior is most marked in the Copy-todeal.II and p4est re-partitioning operations. (The scalability of the latter has been independently demonstrated in Burstedde et al [2010]. )…”

mentioning

confidence: 95%

Algorithms and data structures for massively parallel generic adaptive finite element codes

Bangerth

Burstedde

Heister

et al. 2011

ACM Trans. Math. Softw.

Self Cite

175

154

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“…We use m(T ) to denote both the (d + 1)L-tuple from (14) and the number given by (15). Let us look at Figure 5 for a short example to motivate this definition of the TMindex.…”

Section: Encoding Of the Tetrahedral Morton Indexmentioning

confidence: 99%

“…Using SFCs on hexahedral meshes is exceptionally fast and scalable [14,28,41]. This fact has not only been exploited in writing simulation codes using hexahedral meshes, but also by approaches that use the hexahedral SFC as an instrument to partition simplices, mapping them into a surrounding cube [2].…”

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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“…Most of the papers focus either on providing scalable implementations [10] or showing that applying AMR to a certain problem does not reduce its precision as compared to a uniform mesh [8]. From the few papers we found [11,12], we extracted the main features of the evolution: (i) it is mostly increasing; (ii) it features regions of sudden increase and regions of constancy; (iii) it features some noise.…”

Section: Working Set Evolution Modelmentioning

confidence: 99%

An RMS for Non-predictably Evolving Applications

Klein

Pérez

2011

2011 IEEE International Conference on Cluster Computing

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Non-predictably evolving applications are applications that change their resource requirements during execution. These applications exist, for example, as a result of using adaptive numeric methods, such as adaptive mesh refinement and adaptive particle methods. Increasing interest is being shown to have such applications acquire resources on the fly. However, current HPC Resource Management Systems (RMSs) only allow a static allocation of resources, which cannot be changed after it started. Therefore, non-predictably evolving applications cannot make efficient use of HPC resources, being forced to make an allocation based on their maximum expected requirements. This paper presents CooRMv2, an RMS which supports efficient scheduling of non-predictably evolving applications. An application can make "pre-allocations" to specify its peak resource usage. The application can then dynamically allocate resources as long as the pre-allocation is not outgrown. Resources which are pre-allocated but not used, can be filled by other applications. Results show that the approach is feasible and leads to a more efficient resource usage.Key-words: resource management; supercomputers; clusters; evolving applications; malleability. Un gestionnaire de ressources pour les applications évolutives Résumé :Les applications évolutives non-prédictibles sont des applications dont les besoins en ressources changent en cours d'exécution. Leurs comportements proviennent, par exemple, des méth-odes numériques adaptatives, comme le raffinement de maillage adaptatif et les méthodes de particules adaptatives. Il y a un intérêt croissant à ce que ce type d'application acquiert des ressources à la volée. Cependant, les gestionnaires de ressources haute performance (HPC) ne supportent qu'une allocation statique des ressources, qui ne peut donc pas changer après le démarrage de l'application. Par conséquence, les applications évolutives non-prédictibles ne peuvent pas utiliser les ressources HPC de manière efficace, car elles sont forcées de faire une demande de ressources en fonction de leurs pics de besoins.Cet article présente CooRMv2, un gestionnaire de ressources qui permet l'ordonnancement efficace d'applications évolutives non-prédictibles. Une application peut faire des « pré-allocations » pour exprimer ses pics de besoins en ressources. Ensuite, l'application peut dynamiquement allouer de manière sûre des ressources tant que la pré-allocation n'est pas dépassée. Les ressources pré-allouées, mais inutilisées, sont à la disposition des autres applications. Les résultats montrent que l'approche est réalisable et que l'utilisation des ressources est améliorée.

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Extreme-Scale AMR

Cited by 76 publications

References 35 publications

Algorithms and data structures for massively parallel generic adaptive finite element codes

Algorithms and data structures for massively parallel generic adaptive finite element codes

A Tetrahedral Space-Filling Curve for Nonconforming Adaptive Meshes

An RMS for Non-predictably Evolving Applications

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