Rapid development and adjoining of transient finite element models

Abstract

Recent advances in high level finite element systems have allowed for the symbolic representation of discretisations and their efficient automated implementation as model source code. This allows for the extremely compact implementation of complex non-linear models in a handful of lines of high level code. In this work we extend the high level finite element FEniCS system to introduce an abstract representation of the temporal discretisation: this enables the similarly rapid development of transient finite element models. Efficiency is achieved via aggressive optimisations that exploit the temporal structure, such as automated pre-assembly and caching of forms, and the robust re-use of matrix factorisations and preconditioner data. The resulting models are as fast or faster than hand-optimised finite element codes. The high level representation of the system remains extremely compact and easily manipulated. This structure is exploited to derive the associated discrete adjoint model automatically, with the adjoint model inheriting the performance advantages of the forward model. Combined, this provides a system for the rapid development of efficient transient models, together with their discrete adjoints.

A new gauge-invariant method for diagnosing eddy diffusivities

“…The equations are discretised in space with a conforming triangle structured mesh with piecewise linear approximation for all fields, a vertex spacing of ∆x = 7.5 km, and implemented using the FEniCS automated code generation system (Logg and Wells, 2010;Logg et al, 2012;Alnaes et al, 2014). The model is discretised in time using a third order Adams-Bashforth scheme with time step size ∆t = 20 mins, using the time-stepping approach detailed in Maddison and Farrell (2014). The equations are integrated for 20,000 days and time averages are taken after this spin-up period for a further 5,000 days.…”

Firedrake-Fluids v0.1: numerical modelling of shallow water flows using an automated solution framework

“…taking advantage of the semi-structured nature of a three-dimensional layered mesh extruded in the vertical, as often employed in ocean/atmospheric applications). This is achieved by interfacing with the PyOP2 parallel unstructured mesh computation framework, which targets the automatically generated code towards specific highperformance computing platforms (Rathgeber et al, 2012;Markall et al, 2013;Luporini et al, 2015). In addition, the enhanced abstraction-based approach employed by Firedrake can also help future-proof models from hardware changes and removes a great deal of effort required by computational scientists to maintain the code base.…”

“…C or Fortran) code for the specific set of equations that need to be solved. This task alone can be highly error-prone, often resulting in sub-optimal code, and can make the efficiency, readability, and longevity of the code base difficult to maintain Farrell et al, 2013;Mortensen et al, 2011;Maddison and Farrell, 2014). Moreover, parallelisation of the code is usually accomplished by introducing explicit calls to parallel programming libraries such as OpenMP or CUDA (Compute Unified Device Architecture).…”

“…These benefits have been realised in numerous applications in the geosciences. For example, the use of the FEniCS framework by Maddison and Farrell (2014) allowed the runtime of their adjoint models to be as small as (or even smaller than) an equivalent model generated and optimised by hand. Also, the extension of FEniCS by Rognes et al (2013) to solve partial differential equations on the sphere permits ocean and atmospheric modPublished by Copernicus Publications on behalf of the European Geosciences Union.…”

“…In contrast to Fluidity, Firedrake-Fluids has been written in the high-level Unified Form Language (UFL) and uses Firedrake to automate the solution process. Currently it is capable of solving the non-linear shallow water equations which are widely used in the ocean modelling community for applications such as tidal turbine dynamics (Divett et al, 2013;Kramer et al, 2014;Martin-Short et al, 2015), array optimisation (Funke et al, 2014), tsunami modelling , flow dynamics over submerged islands (Lloyd and Stansby, 1997), and dam breaching and flooding (Capart and Young, 1998). In addition to the core model, FiredrakeFluids offers upwind stabilisation methods, a variety of diagnostic fields, and the Smagorinsky large-eddy simulation (LES) model (Smagorinsky, 1963) for the parameterisation of turbulence.…”