FIREDRAKE '17: THE FIREDRAKE USER AND DEVELOPER WORKSHOP
PROGRAM FOR MONDAY, MARCH 27TH
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10:50-11:10Coffee Break
12:30-13:30Lunch Break
13:30-15:10 Session 3
13:30
The anatomy of the dragon: the concept and design of Firedrake
SPEAKER: David Ham

ABSTRACT. In this presentation I will cover the core ideas underlying the design and implementation of the Firedrake project. I will discuss the mathematical abstractions and their implementation, and introduce the further developments that this has facilitated.

13:55
Compatible finite element methods for numerical weather prediction
SPEAKER: Colin Cotter

ABSTRACT. I will explain what compatible finite element methods are, and why we want to use them in numerical weather prediction models. Then I will explain the special features in Firedrake that enable us to use these methods, and some of the computational science challenges. I will also show some of the latest results using Gusto, the Firedrake Dynamical Core library.

14:20
Variational modelling of water waves and applications to fluid-structure interaction problems
SPEAKER: Onno Bokhove

ABSTRACT. We report on the mathematical and numerical modelling of water waves and their impact on structures. As a first example, we consider the motion of ships and wave-energy devices in (non)linear water waves. A second example concerns waves interacting with the mast of an offshore wind turbine. In both problems, the modelling of the waves is based on a classical potential-flow theory, which assumes an inviscid, incompressible and irrotational fluid, and a variational approach is used so as to ensure zero numerical damping (essential for the propagation of waves). A linear model of the fluid-structure interaction is developed by considering dynamics linearised around a steady state. The model derived is a coupled fluid-structure interaction system that describes the motion of the water waves, the ship or solid structure, and the coupling between them. The system of evolution equations comprises the classical potential-flow water-wave equations (for incompressible and irrotational waves), and a set of equations describing the dynamics of either the ship or an elastic beam (modelled with linear elasticity). In the wave-ship system, we also impose a physical restriction on the water height under the ship, which is enforced through an inequality constraint and by employing a Lagrange multiplier.

The coupled models are solved numerically in Firedrake. In particular, a Galerkin finite element method is used, with continuous linear Lagrange polynomials in space and (dis)continuous 2nd-order approximation in time, yielding the Stormer-Verlet symplectic scheme. The numerical results confirm conservation of total mass and energy, with balanced energy exchange between the subsystems. Future work aims to extend our models to the nonlinear regime, and to investigate the effect of breaking waves on ships and offshore structures.

14:45
Non-dissipative discontinuous Galerkin FEM for internal wave attractors
SPEAKER: Will Booker

ABSTRACT. Confined internal waves can be modeled with a Hamiltonian formulation, that is the motion of the fluid is governed by an energy functional and a (generalised) Poisson bracket. This Poisson bracket embeds the conservation laws and symmetries of the equations of motion.

In this talk we consider a computational approach which models the action for a compressible fluid with a stratified background density. The Poisson bracket is also a weak form that can be written in UFL and then discretised with a discontinuous finite element method. A numerical flux is chosen such that we preserve the underlying Hamiltonian structure. The resulting discrete scheme exactly maintains the conservation properties of the continuous Hamiltonian form.

An incompressible model can then be derived by taking the zero Mach number limit of the compressible model. By preserving the Hamiltonian dynamics at a discrete level, we obtain a stable pressure approximation without satisfying the inf-sup condition.

Firedrake simulations are presented showing the formation of wave attractors.

15:10-15:45Coffee Break
15:45-17:00 Session 4
15:45
Code generation for finite element problems in Firedrake

ABSTRACT. The ability to formulate finite element discretisations and the weak form of partial differential equations using a high-level, mathematical notation is an integral part of the Firedrake system. The form compiler then takes this high-level problem specification in UFL and generates efficient low-level C kernels that carry out the finite element assembly.

The first part of this talk is a brief walkthrough of what happens to a form inside TSFC, the new form compiler employed in Firedrake. The second part highlights some of the recent advancements in Firedrake’s code generation technology, such as efficient form compilation for “complicated” forms and for non-affine (including higher-order) geometries, automatic optimisations for finite element assembly kernels, and support for spectral elements with automatic sum factorisation of tensor product elements.

16:10
Wind-Driven Gyres in Firedrake

ABSTRACT. The planetary scale dynamics in the world’s oceans are often idealized using one-layer models such as the Rotating Shallow Water (SW) and Quasi-Geostrophic (QG) models. These relatively simple models have proved fruitful in explaining qualitatively that western intensification occurs in the ocean gyres due to wind-stress from above and dissipative effects, such as bottom drag or lateral viscosity. Excluding the fact that these models neglect stratification, the major weakness is that it requires parameterizing unresolved scales, and it is not known how to do this in a self-consistent manner.

Recently, we have developed code in Firedrake to calculate the steady wind-driven gyre for both SW and QG models that can include a) bottom drag, b) lateral viscosity and c) nonlinear effects. This approach has several advantages: 1) it is more accurate than the asymptotic solutions that are usually presented, 2) one can easily play with different geometries and 3) it could be used to test different parameterizations.

Furthermore, we have also developed code that will compute the basin modes that can exist in the presence of an wind-driven gyre. This requires solving a linear eigenvalue problem using SLEPc. Both models permit Rossby basin modes but the SW model also allows for inertia-gravity basin modes. Rossby modes are to be important since they contribute to the western intensification and therefore the structure of the current. This code then allows us to easily compare of how the basin modes depend on the structure of the gyre.

16:35
Thetis: Three dimensional baroclinic ocean model
SPEAKER: Tuomas Karna

ABSTRACT. Numerical modeling of coastal ocean systems is challenging due to a wide range of relevant time and length scales, and complex coastal topography. Using an unstructured computational grid is advantageous for capturing the coastal topography (coastlines, islands, channels, estuaries etc.) However, existing unstructured ocean models tend to be diffusive, making it difficult to simulate some key features of coastal systems, such as sharp density fronts in estuaries and river plumes.

We present a novel 3D ocean model, Thetis (http://thetisproject.org/), implemented on Firedrake. Thetis solves the Navier-Stokes equations with hydrostatic and Boussinesq approximations. The solver employs mode-splitting, i.e. the fast propagating surface gravity waves are solved in a 2D domain, coupled with the slowly-varying 3D dynamics. The solver uses P1DG function space and slope limiters, which guarantees conservation and positivity of tracers and yields a 2nd order accuracy.

We present the model formulation and demonstrate its performance and accuracy with several ocean modeling benchmark test cases.