PRISM Co-I Matthew Piggott has been awarded an ARCHER2 eCSE grant to fund a Research Associate (Joe Wallwork) for one year from 1st August 2021 to 31st July 2022. The project is entitled “Goal-Oriented Mesh Adaptation for Firedrake” and will make this advanced discretisation approach available to the Firedrake user community. Goal-oriented mesh adaptation seeks discretisations of the spatial and temporal domain which minimise the error accrued in evaluating a user-provided output diagnostic (such as the power output of a tidal turbine farm or the drag on an aeroplane wing), whilst maintaining low computational resource requirements.

Goal-Oriented Mesh Adaptation for Firedrake

Joseph G. Wallwork and Matthew D. Piggott

Outline

This work builds upon recent metric-based mesh adaptation development in PETSc and Firedrake over the past few years. A major advantage of the metric-based framework is its ability to control the orientation and shape of elements in an adapted mesh, as well as their size. This can be particularly advantageous for advection-dominated problems with a strong directional dependence, such as tidal flows and tsunami propagation.

Thus far, metric-based mesh adaptation in PETSc and Firedrake has been driven by Pragmatic. One of the central objectives of this work is to replace this with the actively developed, open source Mmg mesh adaptation toolkit, in collaboration with researchers at the University of Bordeaux.

This project will also bring new functionality to Firedrake’s user community: goal-oriented mesh adaptation based upon the metric-based framework. Whilst classical mesh adaptation methods typically require a great deal of experience and an in-depth understanding of both the methods and the problem at hand, the goal-oriented approach goes some way to automate the process of generating a suitable discretisation for a particular application. The main two ingredients are a (finite element) PDE solver and a diagnostic quantity of interest (QoI); the output is a mesh or sequence of meshes which admit a finite element solution which minimise the error in the QoI, whilst maintaining a low overall DOF count. Such an adaptation approach is desirable in the many computational science and engineering applications which have a clear QoI. Consider, for example, the drag on an aeroplane wing, the power output of a tidal farm, or the inundation of a coastal settlement due to a tsunami.

Project objectives

• Provide metric-based mesh adaptation functionality to PETSc by coupling its DMPlex unstructured mesh model with Mmg.
• Provide metric-based mesh adaptation functionality to Firedrake via the coupling to PETSc’s DMPlex module.
• Provide goal-oriented mesh adaptation functionality to the community of Firedrake users by creating a new dedicated module. The new module will be extensively documented.
• Perform benchmarking using standard test cases and write training material so that students and researchers can benefit from the research outputs.

Alignment with PRISM strategy

Retention of key staff: This project provides bridge funding for Joe Wallwork. He recently submitted a PhD thesis titled ‘Mesh adaptation and adjoint methods for finite element coastal ocean modelling’, which provides the proof-of-concept implementation of goal-oriented mesh adaptation methods in Firedrake which will act as a basis for this work.

Collaboration with other PRISM projects: This work will be in close collaboration with the Firedrake development team. It will benefit from Firedrake’s improved I/O due to the currently funded ARCHER2 eCSE project led by PRISM Co-I Dr David Ham.

Longer-term research: Users of the (Firedrake-based) Thetis coastal ocean model will benefit from this project, as well as the general Firedrake community. Effective discretisation choice is particularly important for coastal ocean modelling problems, which are typically multi-scale in nature.

International networking: This work will be performed in collaboration with Dr Nicolas Barral and co-workers at the University of Bordeaux. It will also receive support from Prof. Matthew Knepley at the State University of New York.