A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake

Abstract

Abstract. We present a generic algorithm for numbering and then efficiently iterating over the data values attached to an extruded mesh. An extruded mesh is formed by replicating an existing mesh, assumed to be unstructured, to form layers of prismatic cells. Applications of extruded meshes include, but are not limited to, the representation of three-dimensional high aspect ratio domains employed by geophysical finite element simulations. These meshes are structured in the extruded direction. The algorithm presented here exploits this structure to avoid the performance penalty traditionally associated with unstructured meshes. We evaluate the implementation of this algorithm in the Firedrake finite element system on a range of low compute intensity operations which constitute worst cases for data layout performance exploration. The experiments show that having structure along the extruded direction enables the cost of the indirect data accesses to be amortized after 10-20 layers as long as the underlying mesh is well ordered. We characterize the resulting spatial and temporal reuse in a representative set of both continuous-Galerkin and discontinuous-Galerkin discretizations. On meshes with realistic numbers of layers the performance achieved is between 70 and 90 % of a theoretical hardware-specific limit.

Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

“…Thetis is implemented using the Firedrake finite element modeling platform (www.firedrakeproject.org; Rathgeber et al, 2016). We have chosen Firedrake because of its flexibility, and support for extruded meshes (McRae et al, 2016;Bercea et al, 2016). Firedrake uses high-level abstractions for describing the weak formulation of partial differential equations, specifically the Unified Form Language (Alnaes et al, 2014), and automated code generation to produce efficient C code (Homolya et al, 2017;Luporini et al, 2017) and just-in-time compilation.…”

The ‘recovered space’ advection scheme for lowest-order compatible finite element methods

“…The numerical implementation of this scheme was performed using the Firedrake software of [19], and relied heavily on the tensor product element functionality on extruded meshes, descriptions of which can be found in [11], [20] and [21].…”

Comparative Study of Finite Element Methods Using the Time-Accuracy-Size(TAS) Spectrum Analysis

“…The test problems will be tested on the various meshes depicted in Figure 6. The FEniCS/Dolfin and deal.II libraries use custom meshing, while Firedrake uses PETSc to manage unstructured meshes [27][28][29], and the hexahedral meshes are generated using the algorithms described in [10,23,33]. All timings will simply measure the finite element assembly and solve steps but not the mesh generation or any other preprocessing steps.…”