Hierarchical hybrid grids: achieving TERAFLOP performance on large scale finite element simulations

Abstract

The design of the hierarchical hybrid grids (HHG) framework is motivated by the desire to achieve high performance on large-scale, parallel, finite element simulations on super computers. In order to realize this goal, careful analysis of the low-level, computationally intensive algorithms used in implementing the library is necessary. This analysis is primarily concerned with identifying and removing bottlenecks that limit the serial performance of multigrid component algorithms such as smoothing and residual error calculation. To aid in this investigation, two metrics have been developed: the balance metric (BM), and the loads per miss metric (LPMM). Each of these metrics makes assumptions about the interaction of various data structures and algorithms with the underlying memory subsystems and processors of the architectures on which they are implemented. Applying these metrics generates performance predictions that can then be compared to measured results to determine the actual characteristics of an algorithm/data structure on a given platform. This information can then be used to increase performance.In this paper, we first present an overview of the HHG framework. Next, we introduce the details of the two performance metrics. These metrics are then applied to three different data structures used to implement a Gauß-Seidel smoothing algorithm. Performance results and an interpretation of the underlying interactions of the data structures with several relevant supercomputing architectures are given. Finally, we present a brief discussion of some performance results of the HHG framework, followed by some concluding remarks.

Robust Solution of Singularly Perturbed Problems Using Multigrid Methods

“…0.24 (5) 0.89 (6) 1.75 (7) 2.53 (7) 8.10 (7) 16.57 (7) 10 −2 0.46 (6) 0.88 (6) 2.00 (7) 4.06 (7) 8.22 (7) 18.91 (8) (9) 21.20 (9) that this pairwise aggregation never fails. On each mesh, the matrix A is formed as in (2.1).…”

A semi-analytic accuracy benchmark for Stokes flow in 3-D spherical mantle convection codes

“…For details on HHG and the mantle convection prototype implementation see e.g. Bergen and Hülsemann (2004), Bergen et al (2005Bergen et al ( , 2006Bergen et al ( , 2007, Gmeiner et al (2015) and Bauer et al (2016Bauer et al ( , 2019. For our tests we implemented the no-slip version of the benchmark that was presented in the previous chapter, i.e.…”

Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

“…Due to (6), the application of this quadrature rule permits to reduce S to a diagonal matrix. Summarizing, if we chooseṼ h = V h and further use the quadrature rule (7) for computing the integrals on the left-hand sides of (5b) and (5c), the semidiscrete scheme is now given by (5a), (5d) and (5e), in combination with the following equations:…”

“…Note that the value N V /s ∈ N corresponds to the number of edges per subdomain. Furthermore, A 1 is symmetric, positive definite and sparse, and A 2 is diagonal with positive diagonal entries (see (6)). In turn, B is a rectangular block matrix of the form…”

“…In order to construct a hierarchical grid, we consider an unstructured coarse partition of Ω into a certain number of triangular elements, and subsequently apply a regular refinement process to each of these coarse elements 1 . The use of this kind of grids has experienced a great development in recent years, due to their potential for solving a considerable class of problems in large scale simulation (see, e.g., [6,14]). Hierarchical grids are also referred to as semi-structured grids, structured multiblock grids or patch-wise structured grids (cf.…”