ABSTRACT. Many advanced modelling problems in biology, engineering and geological applications can be described by partial differential equations (PDEs) posed on and coupled through domains of possibly different topological dimensionality. A prominent use case are flow and transport problems in porous media when large-scale networks of fractures and channels are modelled as 2D or 1D geometries embedded into a 3D bulk domain. Another important example is the modeling of cell motility where reaction-diffusion systems on the cell membrane and inner cell are coupled to describe the active reorganization of the cytoskeleton. But with complex lower-dimensional and possibly evolving geometries, traditional PDE discretization technologies are severely limited by their strong requirements on the domain discretization.

In this talk, we present the cut finite element (CutFEM) framework --- and its FEniCS-based realization in libcutfem --- as one possible, versatile and unified approach to discretize coupled PDE systems on complicated domains, with a particular emphasis on solving PDEs on embedded manifolds (of codimension $1$ and $2$) .

Along with presentation we will give a number of numerical examples which illustrate the theoretical properties of the framework as well its applicability to a wide range of a complex modelling problem including PDEs on embedded 1D/2D domains coupled to their 3D ambient space.

ABSTRACT. In this presentation, we will give an overview over the features and capabilities of LibCutFEM a cut finite element library based on FEniCS. LibCutFEM extends FEniCS with algorithms to compute unfitted finite element problems in 2D and 3D in level set based geometries. The library can be used to solve fictitious domain and multi-physics problems consisting of multiple materials in different domains as well as surface-bulk coupling problems. In this presentation, we will first briefly introduce the basic algorithms employed in the library, give an example of its use and then demonstrate the capabilities of LibCutFEM for a range of applications including three field Stokes problems and elastodynamic shell computations.

ABSTRACT. Multiphysics problems couple together a set of differential equations to be solved simultaneously. A common example is Stokes' equations for low Reynolds number fluid dynamics.

We have developed an enhanced version of dolfin::SystemAssembler which can assemble a set of forms together with boundary conditions, into a corresponding set of matrices and right hand side vectors. We demonstrate the scaling and memory usage on problems scaling orders of magnitude in size, including on HPC systems.

ABSTRACT. In recent years, finite element methods have gained popularity in the numerical weather prediction community due to their ability to deal with non-orthogonal meshes and high order discretisations. Compatible (or mimetic) spaces offer some appealing features for models such as stable discretisations of pressure gradients and geostrophic balance. However, they present challenges in solving the elliptic system resulting from implicit timestepping schemes. The high aspect ratio domains typical in these problems require us to treat the "vertical" direction specially when designing a solver if we want to achieve mesh-independent iteration counts.

In this talk I will discuss work in Firedrake on developing custom geometric multigrid solvers for these problems and present an analysis of the resulting performance, solving problems with up to two billion degrees of freedom.

ABSTRACT. Simulating ocean flows is challenging due to a wide spectrum of relevant time and length scales. Ocean models have traditionally been implemented in low-level programming languages (e.g. FORTRAN) and designed specifically for certain applications (e.g. global ocean circulation), rendering them rigid and fragile, i.e. difficult to maintain and update. Here we present a novel ocean circulation model, called Thetis, built upon the Firedrake finite element modeling framework \citep{rathgeber2015}.

Firedrake supports extruded meshes that have become customary in ocean modeling: The mesh is unstructured in the horizontal but structured in the vertical direction, allowing separation of horizontal and vertical dynamics. Thanks to the high-level abstractions and automated code generation, the model formulation can be adapted to each application without sacrificing computational efficiency, and it becomes easier to support different HPC hardware. In addition Thetis will support adjoint modeling via dolfin-adjoint \citep{farrell2013} and mesh adaptivity (once available in Firedrake).

Thetis solves the three dimensional Navier-Stokes equations with hydrostatic and Boussinesq approximations. In this presentation we use linear discontinuous Galerkin (DG) elements as well as a mimetic Raviart-Thomas-DG pair. We present model performance in baroclinic test cases and an idealized estuary simulation which includes a turbulence closure model.

ABSTRACT. In many soft-tissue biomechanics simulations the material parameters used in the definition of the hyperelastic energy density functional often have a significant degree of uncertainty associated with them. In a clinical environment, where safety-critical decisions must be made based on the output of simulations, being able to propagate and visualise this uncertainty is of importance.

To propagate uncertainty we recast the the geometrically non-linear Mooney-Rivlin hyperelastic model as a stochastic PDE with random coefficients. We advocate the solution of this non-linear stochastic problem with what we call \textit{partially-intrusive} Monte-Carlo methods. These methods only use the output of the forward model and sensitivity information (tangent linear models derived from UFL expressions) [1] and polynomial chaos expansion (PCE) techniques [2,3] to greatly improve convergence.

We implement our forward and tangent linear model solvers using DOLFIN [4] and we use chaospy [5] to generate various stochastic objects. We then use ipyparallel and mpi4py to massively parallelise individual forward model runs across a cluster.

We compare the results of our method with simple Monte-Carlo methods. By using sensitivity information we demonstrate that computational workload can be reduced by one order of magnitude over commonly used schemes.