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13:00-13:45Welcome coffee.
13:45-15:15 Session 1
Location: Huxley 311
SPEAKER: David Ham
FInAT is not a Tabulator, Or, Yet another way to break a form compiler
SPEAKER: Robert Kirby

ABSTRACT. FIAT (FInite element Automatic Tabulator) and FFC (FEniCS Form Compiler) have made a very effective combination for deploying a wide range of finite elements for variational forms. However, a decade of use has helped to uncover many limitations in the technology. Many such limitations be summarized by saying that FIAT does not expose sufficient structure for a form compiler to make optimal algorithmic decisions. For example, overhauling the current tool chain to enable tensor product elements, nonstandard transformations for Argyris elements, or optimal complexity Bernstein polynomial algorithms would be a major undertaking.

As a forward-looking solution to increasing the power of our automated tools, we propose FInAT, which unlike FIAT, is not a tabulator. Instead, it constructs abstract syntax for fundamental finite element algorithms and can serve as a single entry point for pullbacks, evaluation algorithms, and many optimizations. We will present an overview of FInAT, some examples, and report on preliminary integration with the COFFEE compiler and Firedrake project.

High Order Cut Finite Element Methods for the Stokes Problem using Fenics Multimesh Features

ABSTRACT. We develop a high order finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on a Nitsche formulation of the interface condition together with a stabilization term. Starting from inf-sup stable spaces on the two meshes, we prove that the resulting composite method is indeed inf-sup stable and as a consequence optimal a priori error estimates hold.

We will also present the latest developments on the multimesh features in Fenics.

Using FEniCS on HPC systems

ABSTRACT. Solving FEM problems at large scale involves different challenges from solving on a laptop or workstation. Typically, we have to read a mesh from file, assemble a matrix and vector, call a solver and finally save the solution to file again. Each of these processes becomes more complex as the size of the problem increases, and poor scaling in any part can affect performance. We discuss the challenges associated with I/O, mesh refinement, dof placement and assembly in DOLFIN, along with some profiling results on Intel and Cray HPC machines. Finally, we investigate the current optimum CPU and memory usage for different problem sizes.

15:15-15:45Coffee Break
15:45-17:00 Session 2
Location: Huxley 311
Reliable and efficient a posteriori error estimation in FEniCS
SPEAKER: Jan Blechta

ABSTRACT. Error estimation is an integral part of numerical approximations of practical engineering problems. Optimally, an error estimator gives a guarateed upper bound on the actual error. One possibility to obtain an estimator of this quality is based on the Prager-Synge equality and the flux reconstruction. This is the step needed to compensate for the fact that gradients of discrete solutions are not usually in such a space where a PDE solution is, especially for conforming methods. For instance, the gradient of the Poisson solution is in $H(\mathrm{div})$ space while this is not true for discrete solutions in general.

Current development aims for providing general tools for local and efficient flux reconstruction. In this talk we will discuss several strategies of implementing it using FEniCS. Usefulness of the approaches will be demonstrated on examples.

Exploiting approximation properties to improve filtering.
SPEAKER: Jennifer Ryan

ABSTRACT. Much work goes into creating effective numerical approximations for modeling different physical phenomena such as gas dynamics, climate change, materials, etc. These approximations are usually tuned to the model. However, an important question to ask is how to improve existing numerical approximations. In this talk, we present a generalized discussion concentrating on a basic concept: exploiting the existing approximation properties of a numerical method. We focus on the discontinuous Galerkin (DG) finite element method and show that by simply rewriting the existing approximation in another form, we can take advantage of the underlying approximation method properties. The discontinuous Galerkin method uses a piecewise polynomial approximation to the variational form of a PDE. It uses polynomials up to degree k for a k+1 order accurate scheme.

The main focus of this talk will be on useful superconvergence. Superconvergence is the phenomena of a method to converge faster than expected. For example, discontinuous Galerkin (DG) methods are known theoretically to have order k+1 convergence. However, at specific points within an element, the method has order 2k+1 convergence. Further, DG has the same order of convergence in a negative-order norm, which is achieved through the superconvergent fluxes. Using this information we can construct solutions with reduced errors. We concentrate on B-spline convolution filters known as Smoothness-Increasing Accuracy-Conserving (SIAC) filters which can improve both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k+1 to 2k+1 and smoothness to k-1. This portion of the talk will discuss the barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering for discontinuous Galerkin approximations. We discuss how this property is underutilized, and how it could help is such areas as filtering for visualization.

17:00-19:00 Session 3: Poster evening
Location: Huxley 217/218
Eulerian model of crystal plasticity: numerical simulations of micropillar compression

ABSTRACT. Considering severe plastic deformation experiments as a motivation, plastic behaviour of crystalline solids is treated as a flow of highly viscous material. We present thermodynamic derivation of the model of rate dependent crystal plasticity including evolution of Cauchy stress.

In the last 25 years, the Crystal Plasticity Finite Element Method (CPFEM) has been extensively studied. Most of the methods that have been developed are structural, formulated in Lagrangian coordinates and based on the virtual work principle. Even for large deformations, the increments of the deformation gradient are computed, what is typical for Lagrangian description.

The finite element discretization of the rate type visco-elasto-plastic models leads to several numerical problems. The discussion of proper choice of elements to satisfy the discrete Ladyzhenskaya, Babuška and Brezzi (LBB) condition, on a velocity-pressure-stress (v, p, S) triple, is given.

We present fully Eulerian finite element study of crystal plasticity. Numerical simulations of micropillar compression of face centred cubic (FCC) crystal consists of 12 slip systems, are reported. The Arbitrary Lagrangian Eulerian (ALE) approach is employed in the sense that we use our Eulerian formulation on a moving mesh which captures the free boundary. Our solver is monolithic. The implementation is done in FEniCS package.

A framework for mesh adaptivity in the context of Bayesian inverse problems using FEniCS

ABSTRACT. Formulating an inverse problem in the Bayesian framework allows one to quantify uncertainty in a mathematically rigorous manner. Therein, prior knowledge of the parameters to be inferred, given data, and models that relate parameters to the data are combined using Bayes' rule to provide a revised posterior estimate for the parameters. In the context of physical problems, the models are often partial differential equations (PDEs), with the parameters to be inferred being unknown spatial random fields or locations of sources that drive the problem. Typically the task of solving PDEs is addressed by numerical methods such as the finite element method, thus requiring a mesh. Adaptively building meshes that help achieve a satisfactory solution is essential in order to reduce computational overhead. The topic of error estimation and adaptivity has been explored extensively for deterministic forward problems and continues to be an active research field. Within FEniCS, goal oriented adaptivity using error estimates obtained by duality techniques is currently feasible.

In the context of Bayesian inversion, one encounters additional complexities during the task of error estimation and adaptivity. The parameters being inferred also implicitly define the PDE or the model. A mesh that may work well for one instance of the parameters may not be suitable for another instance. It is thus desirable to obtain a mesh that can reduce the numerical error to a specified tolerance either for all possible instances of the parameters--or, more practically, for parameter values in high-probability regions of the posterior. We formalize this problem by developing posterior-focused error estimates and adaptivity criteria that build on adaptive procedures for deterministic inverse problems. Our approach will combine state of the art statistical inference tools being actively developed by the Uncertainty quantification group at MIT, with mesh adaptivity capabilities available in FEniCS.

Variational modeling of fluid-structure interaction with a free surface
SPEAKER: Tomasz Salwa

ABSTRACT. We consider a model of water waves interacting with a monopile offshore wind turbine fixed to the seabed. We start from a variational principle to describe the full coupled system in both space and time. The water waves are described by nonlinear potential flow with a dynamic free surface and the monopile is represented by a linear elastic beam. A (dis)continuous Galerkin Finite Element Method (DCGFEM) is used, continuous in space and discontinuous in time. A direct discretisation of the variational principle of the coupled system is investigated to yield a stable and compatible numerical scheme. The linear system is derived and discretised first, based on the linearised Luke's variational principle for water waves combined with linear variational theory for the elastic beam. Then the coupled system is solved numerically using the Firedrake package to automate the FEM solution. The nonlinear system is challenging to solve because the domain is moving. This is left as a future work.

A FEniCS-based framework for mesh movement

ABSTRACT. Evolving domain boundaries and varying solution length scales are important characteristics of many fluid dynamics systems, including those involving ice melt/refreezing, sediment scour/deposition, and wear in pipes. The accurate and efficient solution of such systems with finite element methods requires the use of meshing techniques that can incorporate large domain boundary deformations and varying resolution while maintaining mesh quality. Mesh movement, or r-adaptivity, methods are ideal in that they are carefully designed to meet these requirements, with the added benefit that they also avoid changing the mesh topology. Mesh movement methods define the mesh in terms of a mapping based on the solution to an (often nonlinear) elliptic PDE, which itself must be discretized and solved numerically, meaning that implementation of individual schemes is typically a labor-intensive process. In this poster, we present the development of a new mesh movement tool in which the mesh movement PDEs are discretized using FEniCS, automatically generating a moving mesh with users providing only (1) a deformed domain, (2) the underlying (fluids) PDE solution they would like to resolve, and (3) a choice of mesh movement scheme. We apply the included mesh movement methods to a number of test cases to highlight their individual strengths and weaknesses.

CutFEM: unfitted finite element methods for multi-physics problems in FEniCS
SPEAKER: Susanne Claus

ABSTRACT. In this presentation, we will give an overview over CutFEM, a novel stabilised unfitted finite element method for multi-physics problems, and its implementation in FEniCS.

In CutFEM, different PDEs are coupled across an interface that intersects a fixed background mesh in an arbitrary manner. Similar to XFEM, intersected elements in the boundary region are enriched to represent jumps and kinks in the solution inside elements. The boundary conditions on the unfitted interface are enforced using Nitsche-type coupling conditions. Nitsche’s method offers a flexible approach to design XFEM methods that is amenable to analysis. Classically, XFEM methods suffer from ill-conditioning if the interface intersects elements close to element nodes leaving only small parts of the element covered by the physical domain. In our method, we overcome this major difficulty, by adding ghost-penalty terms to the variational formulation over the band of elements that are cut by the interface.

In this presentation, we will give details of how this CutFEM approach combined with Nitsche's method is implemented in FEniCS. We will show computational results for a range of problems both in the bulk and on surfaces with increasing complexity including high contrast multi-physics problems.

A mathematical model of pulmonary blood volume based on a heart rate variability model

ABSTRACT. Cardiovascular and pulmonary perfusion parameters, such as pulmonary blood volume, are some of the most important indicators to diagnose cardiopulmonary disease and they are used in fluid and oxygen therapy. In this work an assessment method of blood volume is proposed. An ordinary differential equation is presented as a simplified model of the pulmonary blood volume, where the source is a waveform of mean arterial pressure in time. Then, since the input to our proposal is a chaotic wave gotten numerically by solving a heart rate variability model, a methodology to solve the problem using the Fast Fourier Transform is described. Through this method is possible to couple a model of baroreflex control to a circulation blood model. As a case of study the new model was simulated with parameters of a healthy subject and a patient with chronic renal failure under two conditions: normal blood volume and lowed blood volume (hypovolaemic). In future work, the simulation of the proposed model will be developed using the FEniCS library.

Modeling Cardiac Arrhythmias of Brugada Syndrome

ABSTRACT. Brugada Syndrome is a hereditary disease characterized by an elevation of the ST segment in the electrocardiographic image of the right bundle branch under sinusoidal rhythm. Patients diagnosed with Brugada Syndrome have a predisposition to malignant ventricular arrhythmias and sudden cardiac arrest. The aim of this work is to explain the generation mechanism of the ST segment elevation in patients with Brugada Syndrome under sinus rhythm. We propose a simplified model based on partial differential equations to simulate the electrocardiographic signals of the three precordial leads where Brugada Syndrome anomalies are manifested. The proposal considers the monodomain model as the mechanism of cell propagation in cardiac tissue, and a phenomenological cell ionic model as the reaction mechanism. The Fenics library will be used to obtain the electrocardiographic signal simulation.

High performance adaptive and predictive finite element computing for turbulent flow and multiphysics
SPEAKER: Johan Jansson

ABSTRACT. We present a general finite element methodology for adaptive and predictive computation of multi-physics and turbulence with an automated realization in the FEniCS-HPC framework [1]. We present challenging applications for prediction of aero- and hydrodynamic forces of a full aircraft [2] and a floating wind turbine platform [3], and turbulent fluid-structure interaction with contact in the human voice apparatus. We estimate the speedup gained from the components of the methodology: adaptivity, slip boundary layer model for high Reynolds number, HPC, etc. to see their relation in contribution to efficiency and cost.

[1] J. Hoffman, J. Jansson, R. Vilela de Abreu, N. C. Degirmenci, N. Jansson, K. Müller, M. Nazarov, J. H. Spühler, "Unicorn: Parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry"q, Computers & Fluids, 2013

[2] J. Hoffman, J. Jansson, N. Jansson, R. Vilela De Abreu, "Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation", Computer Methods in Applied Mechanics and Engineering, 2015

[3] J. Jansson, V. Nava, G. Aguirre, R. Vilela de Abreu, M. Sánchez, G. Pérez, J. Hoffman, J. L. Villate, "Estimation of hydrodynamic viscous characteristics of an offshore wind platform using adaptive finite element simulations", 1st prize for best poster at Bilbao Marine Energy Week, 2015

The optimisation of tidal turbine farms represented as a turbine density function

ABSTRACT. In the design of tidal turbine farms an important question is micro-siting, i.e. the placement of individual turbines within the farm, which is complicated by the fact that each of the turbines interacts with the flow and will affect the performance of other turbines. In principle, all different configurations could be evaluated separately by running a hydrodynamic model for each of them, but due to the large number of possibilities this quickly becomes computationally infeasible even for a small number of turbines. In Funke et al. (2014) a new optimisation strategy for tidal farms is proposed that relies on gradient information, i.e. the sensitivity of the outcomes with respect to the turbine positions. For hydrodynamic models written in (Py)Dolfin this gradient information can be computed through dolfin-adjoint. The micro-siting problem can then be reformulated as a PDE-constrained optimisation problem solved using gradient-based optimisation algorithms. This optimisation strategy for turbine farms has been implemented in OpenTidalFarm, built using the FEniCS framework in combination with dolfin-adjoint.

In this work, the same strategy is followed, but instead of optimising for the positions of individual turbines, we optimise for a turbine density function which indicates the concentration of turbines that should be placed in each area. Although the model gives a less accurate representation of the flow through the farm, a major advantage is that it requires far less mesh resolution as turbines are no longer individually resolved. This makes it feasible to extend the model to larger areas, which is required to reliably study the influence of a farm, or multiple farms, on the large scale tidal flow with boundary conditions sufficiently far removed.

Another advantage of optimising for a turbine density, is that it can be used to determine the optimal *number* of turbines. When optimising the positions of individual turbines, the number of turbines needs to be fixed beforehand. Here however we optimise for all possible turbine densities simultaneously, and determine the number of turbines by simply integrating the optimal density function. Additionally, it becomes possible to enforce much more complex constraints on where turbines are allowed to be placed. Previously, it would only be possible to constrain the turbines to convex areas, and the number of turbines in each separate area would be fixed. In realistic scenarios however, bathymetric constraints for instance may introduce very complex and non-connected areas where turbines can be installed. In this way it also becomes possible to optimise multiple farm sites simultaneously.

We will demonstrate the ability of the method to predict the optimal number of turbines by comparing the results with the previous individual turbine approach run for a different number of turbines. We will also present a large-scale example where multiple sites are optimised simultaneously.

References: S. W. Funke, P. E. Farrell, and M. D. Piggott, “Tidal turbine array optimisation using the adjoint approach”, Renewable Energy, vol. 63, pp. 658–673, 2014.

Micromagnetics with FEniCS

ABSTRACT. The knowledge of the detailed magnetic behavior of small particles is essential in many industrial applications. Hence, a reliable, fast and easily operable code to calculate this behavior with respect to different external influences is of great importance.

We present how FEniCS can help to describe important magnetic effects within nanoparticles in a micromagnetic framework. On the one hand we want to show how the recently developed FEniCS based FEM software magnum.fe can be used very easily within this framework, and on the other hand we want to pick some core elements of the implementation in more detail and present them for a deeper understanding.

PERMON, a new software toolbox for massively parallel computation of real world problems

ABSTRACT. Most engineering problems may be mathematically described by partial differential equations. We discretize them mostly with the popular Finite Element Method (FEM). However, problems that can be expressed in a form of variational inequalities, such as contact problems, naturally lead to quadratic programming problems (QPs). Such problems also arise in applications like least-squares regression, data fitting, data mining, support vector machines, control systems and many others.

In case of large problems solved on supercomputers, domain decomposition methods (DDM) come into play. These mathematical methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains (partitions, substructures) and iterate to coordinate the solution between adjacent subdomains. The subdomain problems are independent, which allows natural parallelization.

Finite Element Tearing and Interconnecting (FETI) methods form a successful subclass of DDM. They belong to non-overlapping methods; this means the subdomains intersect only on their interface. They are based on blending iterative and direct solvers. FETI methods allow highly accurate computations scaling up to tens of thousands of processors.

Almost any engineering problem can be solved by commercially available software packages today. In some cases, due to their limitations, a problem has to be adapted to the tool instead of adapting the tool to the problem. In addition, results may reflect less accurately the physical phenomena in question. Another issue is that it takes a very long time before the state-of-the-art techniques and algorithms are implemented into such packages. It is most painful when porting solvers to HPC platforms is needed. Both these issues lead us to establish the PERMON toolbox.

PERMON is an emerging set of tools for Parallel, Efficient, Robust, Modular, Object-oriented, Numerical simulations. It makes use of theoretical results in advanced discretization techniques, quadratic programming algorithms, and domain decomposition methods. It incorporates our own codes, and makes use of renowned open source libraries.

The core solver layer of PERMON depends on PETSc and uses its coding style. It is formed by these modules: PermonQP for solution of linear systems and quadratic programming problems, PermonFLLOP (FETI Light Layer on Top of PETSc) which is an extension of PermonQP for domain decomposition methods of the FETI type, and PermonIneq adding algorithms for inequality constrained QPs.

Other layers of PERMON include application-specific solver modules such as PermonPlasticity, discretization tools such as PermonCube, interfaces with external discretization software and support tools. We currently focus mainly on the following applications. Engineering applications include mainly structural mechanics – linear elasticity, contact problems with friction, elasto-plasticity, shape optimization. Altruistic applications include medical imaging, ice-sheet melting modelling, and climate changes modelling.

One of great problems is efficient parallel input data generation for large and complicated geometries, suitable for FETI methods. For this purpose we would like to make an interface between the FEniCS library and the PermonFLLOP module, and use FEniCS for parallel generation of input data (e.g. the stiffness matrix, the matrix of the rigid body modes, the mapping vector from local degrees of freedom to global ones and others) on mesh imported from selected commercial software packages.

Parameter-robust discretization and preconditioning of Biot's consolidation model
SPEAKER: Jeonghun Lee

ABSTRACT. In this work, we discuss parameter-robust finite element discretization and its preconditioning for Biot's consolidation model in poroelasticity. A new three-field formulation of the model is proposed. In the new formulation, we can achieve a finite element method and its block diagonal preconditioner for the model, which are robust to various model parameters.

Flexible and efficient mesh management in Firedrake using PETSc DMPlex
SPEAKER: Michael Lange

ABSTRACT. The use of composable abstractions allows the application of new and established algorithms to a wide range of problems while automatically inheriting the benefits of well-known performance optimisations. This work highlights the composition of the PETSc DMPlex domain topology abstraction with the Firedrake automated finite element system to create a PDE solving environment that combines expressiveness, flexibility and high performance. We describe how Firedrake utilises DMPlex to provide the indirection maps required for finite element assembly, while supporting various mesh input formats and runtime domain decomposition. In particular, we describe how DMPlex and its accompanying data structures allow the generic creation of user-defined discretisations, while utilising data layout optimisations that improve cache coherency and ensure overlapped communication during assembly computation.

Mathematical and numerical modelling of variational water waves generated from a wave maker

ABSTRACT. Offshore platforms, wind turbines or ships must be designed to resist the load and stress applied by the waves, whose structure is complex due to non-linearity and the dynamic free surface between water and air. The formulation and the simulation of an accurate mathematical and numerical water wave model would enable us to estimate this load and stress on these structures. In order to test against experiments in a wave basin, this model must also include wave makers to generate the waves.

The linear shallow water equations are derived from a variational principle, in a horizontal plane with a piston wave maker on one side. The model is solved numerically on Firedrake with a space-time Finite Element Method, by using a compatible and consistent space-time discrete variational principle, and the numerical results are compared against a semi-analytical exact standing wave solution. The model is then extended to the non-linear shallow water equations, where the wave maker movement is still linear to keep a fixed domain.

Future directions of this work include extending this model to three dimensions, where the challenge is the unknown time dependent boundary condition due to the free surface of the waves, which requires a moving mesh or coordinates transformation. A second set of wave makers, bottom topography and a beach will also be included to the domain in order to fit the experimental wave basin.

Thermo-Hydro-Mechanical Simulation in Porous Media with FEniCS
SPEAKER: Chao Zhang

ABSTRACT. In this project, we present a fully coupled solver for thermo-hydro-mechanical simulation in porous media. In this multi-physics problem, we are interested in fluid pressure, fluid velocity, solid skeleton displacement, and temperature variation. The model in our study is implemented in FEniCS, with mixed finite element spaces. Specifically, we use Brezzi-Douglas-Marini (BDM) function space and Discontinuous Galerkin (DG) function space for the fluid velocity and pressure field, respectively, to ensure local mass conservation. As the heat transfer process in porous media is usually dominated by advection, the streamline upwind Petrov-Galerkin (SUPG) method is used to stabilize the solution. In addition, a standard Continuous Galerkin interpolant is used for the displacement field.

A wide range of benchmark problems are used to test the performance of the developed solver. Our strategy is to test the interaction between each two of the physical mechanisms first, before going to the final verification of the fully coupled problem. We start with the hydro-mechanical part by simulating the problem of Mandel (1953) and Cryer (1963). While a benchmark problem from Kolditz et al. (2014) is exploited for the thermo-mechanical verification, we use another case from Al-Niami and Rushton (1977) to test the thermo-hydro coupling. Finally, a hypothetical three field problem from Kolditz et al. (2014) is used for the thermo-hydro-mechanical coupling. In all cases, our numerical results are in good agreement with the analytical solutions, which validates our solver and also further demonstrates the applicability of FEniCS in multi-physics applications.

With the resulting solver, the fluid injection/extraction process in a geothermal reservoir is studied. In contrast to TOUGH2 (Pruess, 1991), which is the most widely used simulator in geothermal industry, our FEniCS-based solver is able to analyse the coupled process during geothermal production. Besides, our model also has the potential to study unconventional reservoirs, where the interaction between fluid flow, solid deformation and energy transfer plays an important role.

References A. Al-Niami and K. Rushton. Analysis of flow against dispersion in porous media. Journal of Hydrology, 33(1):87–97, 1977. C. Cryer. A comparison of the three-dimensional consolidation theories of biot and terzaghi. The Quarterly Journal of Mechanics and Applied Mathematics, 16(4):401–412, 1963. O. Kolditz, H. Shao, W. Wang, and S. Bauer. Thermo-Hydro-Mechanical-Chemical Processes in Fractured Porous Media: Modelling and Benchmarking: Closed-Form Solutions. Springer, 2014. 287–299, 1953. J. Mandel. Consolidation des sols. Geotechnique, 3(7):287–299, 1953. K. Pruess. Tough2: A general-purpose numerical simulator for multiphase fluid and heat flow. 1991.

Diderot for Finite Element Data
SPEAKER: Charisee Chiw

ABSTRACT. Diderot is a language designed for efficient analysis and visualization of multi-dimensional scientific images. It supports a high-level mathematical model of computation based on continuous tensor fields. By supporting high-level mathematical abstractions in a familiar notation, Diderot makes common image analysis and visualization algorithms easier to implement. Diderot also makes it easy to take advantage of the extensive parallelism present in these algorithms without forcing the programmer to learn the low-level details of the target platform. Our goal is to enhance Diderot by extending the computation model to include Finite Element Data. This will allow users to visualize their results, such as the boundary of a domain, blood vessels, and energy absorption. The first step is to teach Diderot how to reconstruct fields given a mesh rather than a grid of voxels, reference element, and basis-functions coefficients. Then we hope to provide the user with the ability to load a FEM file with a simple surface level operator that can be easily integrated into Diderot code.

Variational FEM for Waves in a Hele-Shaw Tank

ABSTRACT. The hydrodynamics of the flow in a vertical Hele-Shaw cell are investigated by a model derived from Luke's variational principle. The variational principle is modified to include linear damping due to the effect of wall friction. A simplified set of equations governing the flow are the potential flow shallow water equations. The effect of surface tension and the influence of an artificially driven wave pump have been added to the equations. The sinusoidal motion of the pump produces waves and enters the model as a volume flux boundary condition.

Simulations for the one-dimensional linear system are carried out using the Finite Element package Firedrake. The function space consists of Continuous Galerkin quadratic polynomials, and a second order symplectic scheme is used for the time-discretisation. Finally, the nonlinear system is solved and the numerical results are compared to wave shapes observed experimentally.

Levelset methods (and XFEM) in FEniCS
SPEAKER: Mischa Jahn

ABSTRACT. At the FEniCS'15 conference, we want to present the current state of our work to use the level set method (LSM) in FEniCS. Following the work of [1,2], we consider stabilization, reinitialization and mass correction aspects. All methods are implemented into a FEniCS level set toolbox which is applied to several examples. On top of this toolbox, we are building an XFEM code basing partly on the PUM library by Nikbaht and Wells [3] to handle problems with moving discontinuities.

[1] S. Gross and A. Reusken. Numerical Methods for Two-phase Incompressible Flows. Springer Series in Computational Mathematics. Springer, 2011. [2] R. Ausas, E. Dari, and G. Buscaglia. A mass-preserving geometry-based reinitialization method for the level set function. Mecanica Computacional, 27:25-27, 2008. [3] M. Nikbakht and G. N. Wells. Automated modelling of evolving discontinuities. Algorithms, 2(3):1008–1030, 2009.

On choosing of the free parameters in SUPG and SOLD methods
SPEAKER: Petr Lukas

ABSTRACT. In the talk we consider the numerical solution of the scalar convection-diffusion-reaction equation. We present new results of an adaptive technique in finite element method based on minimizing a functional called error indicator. It is possible to enrich this indicator by other terms, which favour less smeared solution to the diffuse one. The suitability of added terms depends on the problem we solve.

The parameter we are changing in the optimization process is currently the parameter tau from SUPG (SDFEM) method and the parameter called epsilon or sigma from the SOLD method which adds diffusion in the crosswind direction. We use several different FE spaces for both parameters.

Tidal turbine array design optimisation with sensitivity analysis
SPEAKER: Dave Culley

ABSTRACT. From conception to construction, many factors weigh into the process by which tidal turbine farms are scoped and designed. Specification of tidal turbine arrays, both in terms of size (number of turbines) and micro-siting (turbine positions) represents a significant technical challenge which critically impacts upon many aspects of the project, from power extracted to project viability.

The behaviour of a tidal turbine is complex and strongly coupled to the flow in which it is placed. Single turbines will be installed in close proximity to form arrays so as to extract power from tidally induced currents on an industrial scale. Turbines in such proximity influence each other via their effect on the flow; which can be highly variable on a small spacial scale. Not only is the flow speed highly variable, but turbines are highly sensitive to it; the power extracted by a turbine is approximately proportional to the cube of the flow speed.

For simple channel geometries and flow conditions, optimal array designs can be divined intuitively and their optimality proven analytically with relative ease. However, as the complexity of the scenario (domain geometry, bathymetry, practical constraints - such as shipping lanes etc.) grows, it becomes increasingly difficult to intuit array designs which fulfil design objectives, whether those include maximising power extracted from the flow, maximising the return on investment on the project or minimising impact on the environment to name a few.

This work is focused on development of OpenTidalFarm, an open source software project which has developed a framework for the automated optimisation of array design.

Turbines are modelled as areas of increased bed friction and are parametrised by realistic power and thrust curves. The non-linear shallow water equations are discretised by the Taylor-Hood method and solved over the domain using the finite-element method on an unstructured mesh. The key benefit of the approach is that OpenTidalFarm is built on FEniCS and dolfin-adjoint which enable easy and computationally cheap access to gradient information via the adjoint approach. This means that the gradient of the power extracted by the turbines can be computed with respect to the individual (x,y) coordinates of each turbine. Thus a gradient based optimisation scheme may be used to optimise the locations of the turbines, such schemes converge in orders of magnitude fewer iterations that do global optimisation schemes. Fewer iterations means that each iteration may be more computationally expensive; meaning that more of the physical processes may be captured and hence each flow calculation is more realistic.

Tidal stream power is a nascent industry and though there are several projects currently in the planning phases, to date no turbine arrays have been installed. As such, meaningful model validation is essentially impossible until after a first few array deployments have been emplaced and flow measurements with turbines in-situ have been taken. Given this uncertainty, it is imperative that the array designer has a good understanding of the models they are using and the limitations they may have. One of the key focuses of this work is an effort to, in some way, quantify uncertainty in the results produced by the model. Sensitivity analyses of objective functionals with respect to model input parameters are easy and computationally inexpensive to produce. This information would enable developers to see, for example, which areas of the domain it is most important to survey most accurately, and would ensure that the areas to which the array is most sensitive can be explored and defined so that uncertainty is removed from those areas.

dolfin-adjoint: automatic adjoint models for FEniCS
SPEAKER: Simon Funke

ABSTRACT. dolfin-adjoint automatically derives parallel, efficient adjoint and tangent linear models from finite-element models written in the FEniCS environment. It also provides high-level tools to solve PDE-constrained optimisation and generalised stability problems. This poster presents an overview of dolfin-adjoint, including examples and recent developments.

Generating a soliton splash through variational modelling and experiments

ABSTRACT. Mathematical modelling of water waves in tanks with wave generators is demonstrated by investigating variational methods asymptotically and numerically. A reduced potential flow water wave model is derived using variational techniques, which is based on the assumptions of waves with small amplitude and large wavelength. This model consists of a set of modified Benney-Luke equations describing the deviation from the still water surface $\eta(x,y,t)$ and the bottom potential $\Phi(x,y,t)$, and include a time-dependent gravitional potential mimicking a removable ``sluice gate''.

The asymptotic model is solved numerically using the automated system Firedrake. In particular, a (dis)continuous Galerkin finite element method is used, together with 2nd- or 3rd-order symplectic integrators for the time discretisation. As a validation, the numerical results are compared to a soliton splash experiment in a long water channel with a contraction at its end, resulting after a sluice gate is removed at a finite time. Future work includes adapting these methods to accommodate nonlinear ship dynamics in modest to heavy seas.

FEM - BEM coupling methods for the magnetic strayfield problem

ABSTRACT. Maxwell's equations describe how electric and magnetic fields are generated and how they interact with matter. Calculation of the magnetic strayfield leads to an open-boundary problem which requires a large bounding box around the magnetic domain if treated with standalone FEM. While the inhomogeneous and perhaps nonlinear magnetic material is well described by FEM, the boundary element method (BEM) can be used to handle the open-boundary more efficiently and accurately.

We present different types of FEM-BEM coupling methods for the magnetic strayfield problem. These methods can be divided into hybrid methods where FEM and BEM equations are solved subsequently, and direct methods where FEM and BEM equations are combined and need to be solved collectively.

The implementation is based on a combination of FEniCS with BEM++, an open-source Galerkin boundary element library that handles Laplace, Helmholtz and Maxwell problems on unbounded domains. The coupling of both libraries can be done via the high-level NumPy interface. One of the presented methods, namely the Fredkin-Koehler method, has been integrated into magnum.fe, a state-of-the-art library for the solution of dynamical micromagnetic problems.

Anisotropic Mesh Adaptation applied to Structural Optimisation and a Timespace Method

ABSTRACT. Anisotropic mesh adaptation can efficiently resolve features with strong directionality, as often seen in the design of minimally compliant structures as well as for convective problems. At FEniCS'14 we demonstrated the application of two dimensional anisotropic mesh adaptation to structural optimisation. We assumed linear elasticity and focused on stress and compliance constrained volume minimisation. This year, we show how three dimensional mesh adaptation can be used for volume constrained compliance minimisation.

We also show results for a viscous fingering problem solved with a timespace method and anisotropic mesh adaptation. The use of a timespace method avoids the interpolation errors and heuristics, which are associated with combining conventional timestepping methods and mesh adaptation. The chaotic nature of viscous fingering presents some problems for the convergence of our non-linear solver. This can be resolved by bounding the timeslab thickness, i.e. a CFL like condition still applies. Ultimately, a 4D finite element package and anisotropic mesh generator will be needed.

FEniCS in Linux Containers
SPEAKER: Garth Wells

ABSTRACT. We present a collection of Docker images for running FEniCS in Linux containers. With one command, a user can launch a lightweight container that provides a consistent environment for using or developing FEniCS. Once the initial image has been fetched, 'FEniCS terminals' can be launched near-instantly. We show through a range of tests that performance within a container is to equal to that on the host system. Moreover, MPI programs can be run from inside the container, and host CPU vectorisation features can be exploited. In practice, container versions of FEniCS will be faster than user installations as the container images can be carefully tuned for performance. Live demonstrations of user and developer container use will be presented. The containers are built and hosted on Docker Hub ().

Mesh-independent gradient-based optimisation

ABSTRACT. We demonstrate mesh-independent convergence for gradient-based PDE constraint optimisation for infinite-dimensional controls in a framework of the dolfin-adjoint based library OpenTidalFarm. Based on Riesz's representation theorem, a functional derivative can be represented in a number of ways, depending on the choice of the underlying Hilbert space. If the gradient information for the optimisation is incorrectly represented by a Hilbert space different from the control space, the iteration number increases with the refinement of the mesh. For correctly represented gradients, the optimisation iteration number is nearly independent from the considered mesh resolution. These results are embedded in the problem of finding the optimal distribution of turbines within a tidal turbine farm area.

Numerical issues in mathematical modelling of float glass process

ABSTRACT. The float glass process (Pilkington process) is the standard industrial scale process for manufacturing flat glass. The first stage of the process is the flow of the glass melt down an inclined plane, and its impact on the tin bath, which makes the process a practical example of a multicomponent physical system. Our objective is to develop a mathematical model for the process, and implement a numerical scheme that would allow us to perform computer simulations of the process.

For the numerical simulations we use a Cahn-Hilliard-Navier-Stokes type model which conceptually belongs to the class of so-called diffuse interface models. These models treat the interface between the components as a thin layer across which the components can mix, and that, among others, automatically take into account the surface tension effects. This allows one to avoid highly specialized and difficult to implement interface tracking methods. The cost to pay is the need to use a very fine spatial resolution in particular at the interface between the components. This naturally leads to the need to use high performance computers, and to the need to fully exploit the parallelization approach to the scientific computing. The numerical solution is implemented in FEniCS.