Architecture-aware $h$-to-$p$ optimisation: spectral/$hp$ element operators for mixed-element meshes

TL;DR

This paper develops optimized spectral element operators for mixed-element meshes on GPUs, improving performance by tailoring implementations to element shape and polynomial order.

Contribution

It introduces new strategies for efficient tensorial operator evaluation on mixed-element meshes, including a novel approach for complex inner product operations.

Findings

Helmholtz operator throughput on tetrahedral elements is at most 2.5 times slower than on hexahedral elements.

Performance benefits depend on element shape, polynomial order, and system architecture.

New evaluation approach maximizes operations using collocation properties of tensorial expansions.

Abstract

We extend earlier international efforts to optimise hexahedral-based spectral element methods on GPUs and vectorised CPUs to mixed element meshes additionally involving prismatic, pyramidic, and tetrahedral shapes using tensorial expansions. We demonstrate that common finite element operators (such as the mass and Helmholtz matrices) benefit from alternative implementation strategies depending on the element shape, choice of polynomial order, and system architecture in order to achieve optimal performance. In addition, we introduce a new approach/interpretation to efficiently evaluate more complex operations involving inner products with the derivative of the expansions as part of the integrand such as the stiffness matrix. This approach seeks to maximise operations using the collocation properties of the nodal tensorial expansion associated with classical quadrature rules. Our GPU performance tests demonstrate that the throughput of the Helmholtz operator on tetrahedral elements is at most 2.5 times slower than on hexahedral elements, despite tetrahedra having a factor of six greater floating-point operations.