A numbering algorithm for finite elements on extruded meshes which avoids the unstructured mesh penalty

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 3D 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 our algorithm 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 characterise the resulting spatial and temporal reuse in a representative set of both continuous-Galerkin and discontinuous-Galerkin discretisations. On meshes with realistic numbers of layers the performance achieved is between 70 % and 90 % of a theoretical hardware-specific limit.

Reduced storage nodal discontinuous Galerkin methods on semi-structured prismatic meshes

“…It is possible to achieve further speedup by taking advantage of the fact that the wedge meshes used in this work are structured in the vertical direction. This can be exploited to increase data locality by utilizing a layer-by-layer marching algorithm and reusing data from the previous layer [37,38].…”

GPU Support for Automatic Generation of Finite-Differences Stencil Kernels

“…Firedrake [16] is an automated system for the solution of partial differential equations using the finite element method. It relies on UFL from the FeniCS Project to enable an expressive specification of PDEs.…”

Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations

“…The lowest order Raviart-Thomas (RT0) space [Raviart and Thomas, 1977] is employed because it ensures element-wise mass conservation. To map the RT0 element onto quadrilateral and extruded hexahedrons, contravariant Piola mapping is used (see [Rognes et al, 2009;Bercea et al, 2016] for further details). The discrete formulations may be assembled into the following block format:…”