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We present a generic algorithm for numbering and then efficiently iteratingover the data values attached to an extruded mesh. An extruded mesh is formedby replicating an existing mesh, assumed to be unstructured, to form layers ofprismatic cells. Applications of extruded meshes include, but are not limitedto, the representation of 3D high aspect ratio domains employed by geophysicalfinite element simulations. These meshes are structured in the extrudeddirection. The algorithm presented here exploits this structure to avoid theperformance penalty traditionally associated with unstructured meshes. Weevaluate the implementation of this algorithm in the Firedrake finite elementsystem on a range of low compute intensity operations which constitute worstcases for data layout performance exploration. The experiments show that havingstructure along the extruded direction enables the cost of the indirect dataaccesses to be amortized after 10-20 layers as long as the underlying mesh iswell-ordered. We characterise the resulting spatial and temporal reuse in arepresentative set of both continuous-Galerkin and discontinuous-Galerkindiscretisations. On meshes with realistic numbers of layers the performanceachieved is between 70% and 90% of a theoretical hardware-specific limit.

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