@misc {26660,
title = {Algorithmic differentation for shape derivatives with PDE constraints},
howpublished = {Berlin, Germany},
year = {2019},
type = {Conference talk},
abstract = {Shape-derivatives combined with the adjoint approach offer an efficient approach for solving shape optimisation problems, but their derivation is error-prone especially for complex PDEs. In this talk, we present an algorithmic differentiation tool that automatically computes shape derivatives by aexploiting the symbolic representation of variational problems in the finite element framework FEniCS. We demonstrate that our approach computes first and second order shape derivatives for a wide range of PDEs and functionals, approaches optimal performance and works naturally in parallel.},
author = {Simon W {Funke} and J{\o}rgen Schartum {Dokken} and Mitusch, Sebastian}
}
@misc {26661,
title = {Algorithmic differentiation for coupled FEniCS and PyTorch models},
year = {2019},
address = {ICCOPT, Berlin, Germany},
abstract = {This poster presents recent advances in dolfin-adjoint, the algorithmic differentiation tool for the finite element framework FEniCS. We will show dolfin-adjoint can be used to differentiated and optimised PDE models in FEniCS that are coupled to neural network models implemented in pyTorch. Our approach allows to compute derivatives with respect to weights in the neural network, and as PDE coefficients such as initial condition, boundary conditions and even the mesh shape.},
author = {Mitusch, Sebastian and Simon W {Funke}}
}
@article {26525,
title = {Automated adjoints of coupled PDE-ODE systems},
journal = {SIAM Journal on Scientific Computing},
volume = {41},
year = {2019},
publisher = {SIAM},
doi = {10.1137/17M1144532},
url = {http://epubs.siam.org/toc/sjoce3/41/3 },
author = {Patrick E. {Farrell} and Johan Elon {Hake} and Simon W {Funke} and Marie E. {Rognes}}
}
@misc {26987,
title = {Compiling finite element variational forms for GPU-based assembly},
howpublished = {FEniCS{\textquoteleft}19, Washington DC, USA},
year = {2019},
abstract = {We present an experimental form compiler for exploring GPU-based algorithms for assembling vectors, matrices, and higher-order tensors from finite element variational forms.Previous studies by Cecka et al. (2010), Markall et al. (2013), and Reguly and Giles (2015) have explored different strategies for using GPUs for finite element assembly, demonstrating the potential rewards and highlighting some of the difficulties in offloading assembly to a GPU. Even though these studies are limited to a few specific cases, mostly related to the Poisson problem, they already indicate that to achieve high performance, the appropriate assembly strategy depends on the problem at hand and the chosen discretisation.By using a form compiler to automatically generate code for GPU-based assembly, we can explore a range of problems based on different variational forms and finite element discretisations. In this way, we aim to get a better picture of the potential benefits and challenges of assembling finite element variational forms on a GPU. Ultimately, the goal is to explore algorithms based on automated code generation that offload entire finite element methods to a GPU, including assembly of vectors and matrices and solution of linear systems.In this talk, we give an exact characterisation of the class of finite element variational forms supported by our compiler, comprising a small subset of the Unified Form Language that is used by FEniCS and Firedrake. Furthermore, we describe a denotational semantics that explains how expressions in the form language are translated to low-level C or CUDA code for performing assembly over a computational mesh. We also present some initial results and discuss the performance of the generated code.},
keywords = {Code translation, GPU, HPC},
author = {James D. {Trotter} and Cai, Xing and Simon W {Funke} and Langguth, Johannes}
}
@article {26585,
title = {dolfin-adjoint 2018.1: automated adjoints for FEniCS and Firedrake},
journal = {Journal of Open Source Software},
volume = {4},
year = {2019},
month = {Jan-06-2019},
pages = {1292},
publisher = {JOSS},
doi = {10.21105/joss.01292},
author = {Mitusch, Sebastian and Simon W {Funke} and J{\o}rgen Schartum {Dokken}}
}
@article {27070,
title = {A multimesh finite element method for the Navier-Stokes equations based on projection methods},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2019},
publisher = {Elsevier},
abstract = {The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche{\textquoteright}s method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable variational formulation. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure correction scheme. For each step in the pressure correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The overall scheme yields expected spatial and temporal convergence rates on the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients on the Turek-Schafer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a channel.},
url = {https://arxiv.org/abs/1912.06392},
author = {J{\o}rgen Schartum {Dokken} and Johansson, August and Massing, Andre and Simon W {Funke}}
}
@misc {26662,
title = {Recent developments in dolfin-adjoint},
howpublished = {FEniCS{\textquoteright}19, Washington DC, USA},
year = {2019},
abstract = {dolfin-adjoint is a python library that enables automatic differentiation and optimization of FEniCS models by deriving and solving the corresponding first and second-order adjoint equations. In the last two years, dolfin-adjoint has been completely rewritten to accommodate the implementation of new features. For instance, we have implemented shape derivatives, which was enabled by a recent extension to UFL [1]. We will present a highlight of the new features in dolfin-adjoint, including deformation vector and strong Dirichlet boundary condition controls. Furthermore, we present the performance of these implementations compared to the theoretical optimum. Lastly, we mention how dolfin-adjoint can be extended to support new operations.[1] David A Ham, Lawrence Mitchell, Alberto Paganini, and Florian Wechsung. Automated shape differentiation in the unified form language. arXiv preprint arXiv:1808.08083, 2018.},
author = {Mitusch, Sebastian and Simon W {Funke} and J{\o}rgen Schartum {Dokken}}
}
@article {26784,
title = {Shape Optimization Using the Finite Element Method on Multiple Meshes with Nitsche Coupling},
journal = {SIAM Journal on Scientific Computing},
volume = {41},
year = {2019},
month = {Feb-01-2019},
pages = {A1923 - A1948},
publisher = {SIAM},
abstract = {An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other typically lacks robustness or is computationally expensive. This paper proposes a different approach, in which the computational domain is represented by multiple, independent meshes. A Nitsche based finite element method is used to weakly enforce continuity over the non-matching mesh interfaces. The optimization is preformed using an iterative gradient method, in which the shape-sensitivities are obtained by employing the Hadamard formulas and the adjoint approach. An optimize-then-discretize approach is chosen due to its independence of the FEM framework. Since the individual meshes may be moved freely, re-meshingor mesh deformations can be entirely avoided in cases where the geometry changes consists of rigid motions or scaling. By this free movement, we obtain robust and computational cheap mesh adaptation for optimization problems even for large domain changes. For general geometry changes, the method can be combined with mesh-deformation or re-meshing techniques to reduce the amount of deformation required. We demonstrate the capabilities of the method on several examples, including the optimal placement of heat emitting wires in a cable to minimize the chance of overheating, the drag minimization in Stokes flow, and the orientation of 25 objects in a Stokes flow.},
issn = {1064-8275},
doi = {10.1137/18M1189208},
url = {https://epubs.siam.org/doi/10.1137/18M1189208https://epubs.siam.org/doi/pdf/10.1137/18M1189208},
author = {J{\o}rgen Schartum {Dokken} and Simon W {Funke} and Johansson, August and Schmidt, Stephan}
}
@article {paper27,
title = {Towards personalized computer simulation of breast cancer treatment: a multi-scale pharmacokinetic and pharmacodynamic model informed by multi-type patient data},
journal = {Cancer Research},
volume = {79},
year = {2019},
month = {05/2019},
pages = {4293-4304},
publisher = {American Association for Cancer Research},
doi = {10.1158/0008-5472.CAN-18-1804},
url = {http://cancerres.aacrjournals.org/content/early/2019/05/22/0008-5472.CAN-18-1804},
author = {Lai, Xiaoran and Geier, Oliver and Fleischer, Thomas and Garred, {\O}ystein and Elin Faye {Borgen} and Simon W {Funke} and Kumar, Surendra and Marie E. {Rognes} and Seierstad, Therese and Boerresen-Dale, Anne-Lise and Vessela N. {Kristensen} and Engebraaten, Olav and Kohn-Luque, Alvaro and Frigessi, Arnoldo}
}
@article {25074,
title = {The trade off between tidal-turbine array yield and environmental impact: a multi-objective optimisation problem},
journal = {Renewable Energy},
volume = {114},
year = {2019},
pages = {390-403},
publisher = {Elsevier},
abstract = {In the drive towards a carbon-free society, tidal energy has the potential to become a valuable part of the UK energy supply. Developments are subject to intense scrutiny, and potential environmental impacts must be assessed. Unfortunately many of these impacts are still poorly understood, including the implications that come with altering the hydrodynamics. Here, methods are proposed to quantify ecological impact and to incorporate its minimisation into the array design process. Four tidal developments in the Pentland Firth are modelled with the array optimisation tool OpenTidalFarm, that designs arrays to generate the maximum possible profit. Maximum entropy modelling is used to create habitat suitability maps for species that respond to changes in bed-shear stress. Changes in habitat suitability caused by an altered tidal regime are assessed. OpenTidalFarm is adapted to simultaneously optimise array design to maximise both this habitat suitability and to maximise the profit of the array. The problem is thus posed as a multi-objective optimisation problem, and a set of Pareto solutions found, allowing trade-offs between these two objectives to be identified. The methods proposed generate array designs that have reduced negative impact, or even positive impact, on the habitat suitability of specific species or habitats of interest.},
doi = {10.1016/j.renene.2019.04.141},
author = {Roan du {Feu} and Simon W {Funke} and Stephan C. {Kramer} and Dave M. {Culley} and Hill, J and B S {Halpern} and Matthew D. {Piggott}}
}
@misc {25862,
title = {Algorithmic differentiation for mixed FEniCS-Tensorflow models},
howpublished = {Oxford, UK},
year = {2018},
type = {FEniCS 2018},
abstract = {In this talk, we present a recent addition to dolfin-adjoint: support for Tensorflow models. Tensorflow is a numerical computation library which is mostly used to conduct machine learning (ML) and deep neural networks research, but which can also be applied in various other domains. With Tensorflow support in dolfin-adjoint, users can now implement mixed PDE-ML models, where the PDEs are solved with FEniCS and the ML functions are computed with Tensorflow. The high-level differentiation capabilities of dolfin-adjoint automatically compute derivatives of these models, which may be used to optimise (train) model parameters from data.The implementation works as follows. During model execution, an algorithmic differentiation (AD) tool records a computation graph. In this graph, nodes represent high-level mathematical operations such as a PDE solve using FEniCS, or a ML function evaluation using Tensorflow. Edges in this graph represent variables, such as finite element functions or tensors, that flow between operations. Further, we implemented basic transfer functions to map tensors in Tensorflow to FEniCS data structures. From this computation graph, dolfin-adjoint derives the associated adjoint graph, in which each node is replaced by its assoiated adjoint. If a graph node represents a PDE solve, the AD capabilities in FEniCS is used to obtain its adjoint. Otherwise, if a node represents a Tensorflow operation, then Tensorflow{\textquoteright}s internal AD tool is used to obtain the adjoint version of that operation.},
author = {Simon W {Funke} and Mitusch, Sebastian}
}
@misc {26262,
title = {Algorithmic Differentiation for Shape Optimization problems with overlapping meshes},
howpublished = {Siam Annual Meeting, Portland, USA},
year = {2018},
author = {J{\o}rgen Schartum {Dokken}},
editor = {Simon W {Funke} and Johansson, August}
}
@misc {26153,
title = {Automated adjoints for finite element models},
howpublished = {EUCCO 2018, Trier, Germany},
year = {2018},
abstract = {Adjoints of partial differential equations (PDEs) play an key role in solving optimizationproblems constrained by physical laws. The adjoint model efficiently computes gradient andHessian information, and hence allows the use of derivative based optimisation algorithms.While deriving the adjoint model associated with a linear stationary PDE model is straightfor-ward, the derivation and implementation of adjoint models for non-linear or time-dependentPDE models is notoriously difficult.In this talk, we solve this problem by automatically deriving adjoint models for finiteelement models. Our approach raises the level of abstraction of algorithmic differentiationfrom the level of individual floating point operations to that of entire systems of differentialequations. For each differential equation, the algorithm analyses and exploits the high-levelmathematical structure inherent in finite element methods to derive its adjoint. We demons-trate that this strategy has advantages over traditional algorithmic differentiation: the adjointmodel is robustly obtained with minimal code changes, yields close-to-optimal performanceand inherits the parallel performance of the forward model.The library dolfin-adjoint implements this idea as an extension to the FEniCS Project.\ Recently, a major update to dolfin-adjoint has been a released. This talk will showcasesome of the new features, including differentiation with respect to Dirichlet boundary condi-tions, automated shape derivatives, and the experimental integration with a machine learningframework. In addition, we show applications where dolfin-adjoint has already been employed.},
author = {Simon W {Funke}}
}
@misc {25873,
title = {High-level abstractions for optimal checkpointing in inversion problems},
howpublished = {Bordeaux},
year = {2018},
publisher = {ISMP2018},
type = {Oral},
address = {Bordeaux},
abstract = {Inversion and PDE-constrained optimization problems often relyon solving the adjoint problem to calculate the gradient of the objectivefunction. This requires storing large amounts of intermediatedata, setting a limit to the largest problem that might be solved witha given amount of memory available. Checkpointing is an approachthat can reduce the amount of memory required by redoing partsof the computation instead of storing intermediate results. TheRevolve checkpointing algorithm offers an optimal schedule thattrades computational cost for smaller memory footprints. By varying the\ number of checkpoints to be stored in memory, the memory footprint of\ the resulting algorithm can be adjusted to fit within a large range of target\ machines. Integrating Revolve into a modern python HPC code and\ combining it with code generation is not straightforward. We present anAPI that makes checkpointing accessible from a DSL-based code generationenvironment along with some initial performance figures with afocus on seismic-imaging applications.\ },
url = {https://ismp2018.sciencesconf.org/},
author = {Kukreja, Navjot and Huckelheim, Jan and Lange, Michael and Louboutin, Mathias and Walther, Andrea and Simon W {Funke} and Gorman, Gerard}
}
@misc {25847,
title = {Ideas on how to combine machine learning with physical simulations},
howpublished = {Oslo, Norway},
year = {2018},
type = {Company lunch talk at Kalkulo},
author = {Simon W {Funke}}
}
@article {25968,
title = {In vivo estimation of elastic heterogeneity in an infarcted human heart},
journal = {Biomechanics and Modeling in Mechanobiology},
volume = {17},
year = {2018},
month = {May-05-2019},
pages = {1317{\textendash}1329},
publisher = {Springer},
address = {Berlin Heidelberg},
abstract = {In myocardial infarction, muscle tissue of the heart is damaged as a result of ceased or severely impaired blood flow. Survivors have an increased risk of further complications, possibly leading to heart failure. Material properties play an important role in determining post-infarction outcome. Due to spatial variation in scarring, material properties can be expected to vary throughout the tissue of a heart after an infarction. In this study we propose a data assimilation technique that can efficiently estimate heterogeneous elastic material properties in a personalized model of cardiac mechanics. The proposed data assimilation is tested on a clinical dataset consisting of regional left ventricular strains and in vivo pressures during atrial systole from a human with a myocardial infarction. Good matches to regional strains are obtained, and simulated equi-biaxial tests are carried out to demonstrate regional heterogeneities in stress{\textendash}strain relationships. A synthetic data test shows a good match of estimated versus ground truth material parameter fields in the presence of no to low levels of noise. This study is the first to apply adjoint-based data assimilation to the important problem of estimating cardiac elastic heterogeneities in 3-D from medical images.},
issn = {1617-7959},
doi = {10.1007/s10237-018-1028-5},
url = {http://link.springer.com/10.1007/s10237-018-1028-5},
author = {Balaban, Gabriel and Finsberg, Henrik and Simon W {Funke} and Trine F. {H{\r a}land} and Hopp, Einar and Sundnes, Joakim and Wall, Samuel and Marie E. {Rognes}}
}
@misc {25849,
title = {MiniBiz: Can we combine machine learning and physical simulations?},
howpublished = {StartupLab, Oslo, Norway},
year = {2018},
author = {Simon W {Funke}}
}
@misc {25867,
title = {Towards Algorithmic Differentiation of shape optimization problems with time-dependent PDE-constraints},
year = {2018},
address = {FEniCS 18, Oxford, UK},
keywords = {Algorithmic Differentiation, FEniCS, pyadjoint, Shape optimization},
author = {J{\o}rgen Schartum {Dokken}},
editor = {Simon W {Funke}}
}
@article {25229,
title = {Variational data assimilation for transient blood flow simulations},
journal = { International Journal for Numerical Methods in Biomedical Engineering},
volume = {35},
year = {2018},
month = {10/2018},
pages = {e3152},
publisher = {John Wiley \& Sons},
abstract = {Several cardiovascular diseases are caused from localised abnormal blood flow\ such as in the case of stenosis or aneurysms. Prevailing theories propose that the development is caused by abnormal wall-shear stress in focused\ areas. \ Computational fluid mechanics have arisen as a promising tool for\ a more precise and quantitative analysis, in particular because the\ anatomy is often readily available even by standard imaging techniques such as magnetic resolution and computed tomography angiography. \ However,\ computational fluid mechanics rely on accurate initial and boundary conditions which\ is difficult to obtain. \ In this paper we address the problem of recovering high resolution information from noisy, low-resolution\ measurements of blood flow using variational data assimilation based on a transient Navier-Stokes model.\ Numerical experiments are performed in both 2D and 3D and with pulsatile flow relevant for physiological flow in cerebral aneurysms.\ The results demonstrate that, with suitable regularisation, the model accurately reconstructs flow, even in the presence of significant noise.\ },
keywords = {adjoint equations, blood flow, Finite element method, Navier-Stokes, optimal control, variational data assimilation},
author = {Simon W {Funke} and Nordaas, Magne and Evju, {\O}yvind and Martin Sandve {Aln{\ae}s} and Mardal, Kent-Andre}
}
@article {25072,
title = {Application of the adjoint approach to optimise the initial conditions of a turbidity current (AdjointTurbidity 1.0)},
journal = {Geoscientic Model Development},
year = {2017},
pages = {1051-1068},
publisher = {Copernicus Publications},
doi = {10.5194/gmd-10-1051-2017},
author = {Sam D. {Parkinson} and Simon W {Funke} and Hill, Jon and Matthew D. {Piggott} and Peter A. {Allison}}
}
@misc {25145,
title = {Automatic Adjoints of Multimesh Finite Element Discretisations},
howpublished = {Atlanta, USA},
year = {2017},
publisher = {SIAM Conference on Computational Science \& Engineering},
abstract = {Many interesting physics-driven optimisation problems include dynamic domains (i.e. domains that change in time). One can think of the fluid-fluid interaction in a mixing process or the fluid-structure interaction of a rotating turbine or propeller. In this talk, we present an approach for solving such dynamic optimisation problems based on a multi-mesh idea: we allow the computational domain to consist of multiple meshes which can independently move. These meshes are then coupled in the variational formulation, typically through Nitsche terms. This approach is particularly promising in a PDE-constrained optimisation setting, since it allows even large movements of the domain without non-differentiable and computationally expensive remeshing steps. We implemented this approach within the FEniCS and dolfin-adjoint projects and demonstrate that it can be combined with a high-degree of automation and code generation. The user specifies the variational formulation including the coupling terms in the domain-specific language UFL, from which assembly code is automatically generated via the FEniCS form compiler. The associated adjoint equations are derived automatically from the UFL description, and solved for a specific forward state and mesh-movements. We give examples of PDE-constrained optimisation problems with dynamic domains that can now be solved in a few dozen lines of Python code.},
author = {Simon W {Funke} and J{\o}rgen Schartum {Dokken} and Johansson, August and Schmidt, Stephan}
}
@article {25249,
title = {cbcbeat: an adjoint-enabled framework for computational cardiac electrophysiology},
journal = {Journal of Open Source Software},
volume = {2},
year = {2017},
month = {06/2017},
publisher = {The Journal of Open Source Software, Open Source Initiative},
doi = {10.21105/joss.00224},
url = {http://dx.doi.org/10.21105/joss.00224},
author = {Marie E. {Rognes} and Patrick Emmet {Farrell} and Simon W {Funke} and Johan Elon {Hake} and Maleckar, Molly}
}
@article {24876,
title = {Hybrid Genetic Deflated Newton Method for Global Optimisation},
journal = {Journal of Computational and Applied Mathematics},
volume = {325},
year = {2017},
pages = {97-112},
publisher = {Elsevier},
abstract = {Optimisation is a basic principle of nature and has a vast variety of applications in research and industry. There is a plurality of different optimisation procedures which exhibit different strengths and weaknesses in computational efficiency and probability of finding the global optimum. Most methods offer a trade-off between these two aspects. This paper proposes a hybrid genetic deflated Newton (HGDN) method to find local and global optima more efficiently than competing methods. The proposed method is a hybrid algorithm which uses a genetic algorithm to explore the parameter domain for regions containing local minima, and a deflated Newton algorithm to efficiently find their exact locations. In each iteration, identified minima are removed using deflation, so that they can not be found again. The genetic algorithm is adapted as follows: every individual of every generation of offspring searches its adjacent space for optima using Newton\’s method; when found, the optimum is removed by deflation, and the individual returns to its starting position. This procedure is repeated until no more optima can be found. The deflation step ensures that the same optimum is not found twice. In the subsequent genetic step, a new generation of offspring is created, using procreation of the fittest. We demonstrate that the proposed method converges to the global optimum, even for small populations. Furthermore, the numerical results show that the HGDN method can improve the number of necessary function and derivative evaluations by orders of magnitude.},
doi = {10.1016/j.cam.2017.04.047},
url = {http://dx.doi.org/10.1016/j.cam.2017.04.047},
author = {Noack, Marcus and Simon W {Funke}}
}
@misc {25358,
title = {Machine learning with expert systems},
howpublished = {Simula Research Laboratory, Norway},
year = {2017},
type = {COMMONS seminar},
abstract = {The sensitivity of computer program outputs to its inputs is a driving component of many algorithms that surround our daily live. One example is machine learning, where sensitivities are used to train the machine to a specific data set. Another example is simulation-based optimisation, where the sensitivities are used to improve physically constrained designs. In this talk, I will present the numerical techniques that is used to compute these sensitivities. We will see that the similar techniques are used both in machine learning (known as back-propagation) and simulation-based optimisation (adjoint methods). Finally, we investigate possibilities to combine simulation-based optimisation and machine learning to leverage the advantages of both approaches.},
author = {Simon W {Funke}}
}
@book {25071,
title = {Mesh dependence in PDE-constrained optimisation},
series = {An Application in Tidal Turbine Array Layouts},
year = {2017},
pages = {111},
publisher = {Springer Research Brief},
organization = {Springer Research Brief},
edition = {1},
address = {Berlin / Heidelberg},
abstract = {This book provides an introduction to PDE-constrained optimisation using finite elements and the adjoint approach. The practical impact of the mathematical insights presented here are demonstrated using the realistic scenario of the optimal placement of marine power turbines, thereby illustrating the real-world relevance of best-practice Hilbert space aware approaches to PDE-constrained optimisation problems.Many optimisation problems that arise in a real-world context are constrained by partial differential equations (PDEs). That is, the system whose configuration is to be optimised follows physical laws given by PDEs. This book describes general Hilbert space formulations of optimisation algorithms, thereby facilitating optimisations whose controls are functions of space. It demonstrates the importance of methods that respect the Hilbert space structure of the problem by analysing the mathematical drawbacks of failing to do so. The approaches considered are illustrated using the optimisation problem arising in tidal array layouts mentioned above.This book will be useful to readers from engineering, computer science, mathematics and physics backgrounds interested in PDE-constrained optimisation and their real-world applications.},
isbn = {978-3-319-59482-8},
url = {http://www.springer.com/us/book/9783319594828},
author = {Schwedes, Tobias and David A. {Ham} and Simon W {Funke} and Matthew D. {Piggott}}
}
@misc {25418,
title = {A new algorithmic differentiation tool (not only) for FEniCS},
year = {2017},
note = {Winner of the FEniCS 2017 Best Poster Competition},
address = {FEniCS 2017 conference, Luxembourg},
abstract = {The derivation and implementation of adjoint models for time-dependent, non-linear PDEs is a challenging task. A common strategy is to an apply algorithmic differentiation tool (AD) which (semi- )automatically derives the adjoint model from the forward model. Specifically for finite-element models, [1] proposed a high-level AD approach, which derives the adjoint by analysing and exploiting the high- level mathematical structure inherent in finite element methods. This idea has shown to provide major benefits compared to traditional low-level AD tools, including near to theoretically optimal performance and natural support of parallelism. However, the high-level AD tool for FEniCS, dolfin-adjoint, lacks important features such as differentiation with respect to Dirichlet boundary conditions and higher-order derivatives. To overcome these limitations, we propose a new algorithmic differentiation tool for FEniCS. The core of this tool is formed by pyadjoint, a generic operator overloading AD tool written in Python. pyadjoint considers the model as a sequence of arbitrary operations with inputs and outputs. This abstraction can be seen as a generalisation of low and high-level AD tools: operations can be individual floating point operations (as for traditional AD tools), entire systems of differential equations (as for high-level AD tools), or a mix of both. The adjoint developer must overload each relevant model function according to the pyadjoint API, and in particular provide implementations for their derivatives. With this information pyadjoint records a tape of model operations at runtime and automatically derives and executes the associated adjoint model. Specifically, the support for adjoint FEniCS model is achieved by overloading of the FEniCS API, in particular the creating of new objects such as Functions, Constants, and overloading operators such as assemble, project and solve.},
url = {http://easychair.org/smart-program/FEniCS{\textquoteright}17/2017-06-12.html$\#$talk:45748},
author = {Mitusch, Sebasitan and Simon W {Funke}}
}
@misc {25452,
title = {Optimization Problems in Dynamical Domains with Unfitted Meshes},
howpublished = {X-DMS 2017, Ume{\r a}, Sweden},
year = {2017},
author = {Simon W {Funke} and J{\o}rgen Schartum {Dokken} and Johansson, August and Schmidt, Stephan}
}
@misc {25471,
title = {The Power of Python in Science and Education},
howpublished = {Fysikerm{\o}tet 2017, Troms{\o}, Norway},
year = {2017},
type = {Plenary lecture},
abstract = {Researchers often translate new ideas into computer programs to test them. Successful ideas will be used in future research, so these programs must be extendable and robustly. The choice of programming language is important as it can have a big impact on how quickly we can test new ideas. So which language should we choose?\ The Python programming language has proven to be surprisingly productive for scientists. Join this talk to learn why - and how Python can make your research and education more effective as well.},
url = {https://docs.google.com/presentation/d/1nIpQfO2aQilM8cnhpMPtIx27IdnlXCH7RRpH-tPf6xA/edit?usp=sharing},
author = {Simon W {Funke}}
}
@article {24986,
title = {Reconstructing wave profiles from inundation data},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {322},
year = {2017},
pages = {167-186},
publisher = {Elsevier},
abstract = {This paper applies variational data assimilation to inundation problemsgoverned by the shallow water equations with wetting and drying. \ Theobjective of the assimilation is to recover an unknown time-varying wave profile at an openocean boundary from inundation observations. This problem is solved with\ derivative-based optimisation and an adjoint wetting and dryingscheme to efficiently compute sensitivity information. The capabilities of this approach aredemonstrated on an idealised sloping beach setup in which the profile of anincoming wave is reconstructed from wet/dry interface observations. The method isrobust against noisy observations if a regularisation term is added to the optimisation objective.Finally, the method is applied to a laboratory experiment of the Hokkaido-Nansei-Oki tsunami,where the wave profile is reconstructed with an error of less than 1\% of the reference wave signal.},
doi = {10.1016/j.cma.2017.04.019},
author = {Simon W {Funke} and Patrick Emmet {Farrell} and Matthew D. {Piggott}}
}
@misc {25419,
title = {Sensitivity Analysis of Cardiac Growth Models},
year = {2017},
address = {FEniCS 2017 conference, Luxembourg},
abstract = {\ Introduction The heart is a dynamic organ capable of changing its shape in response to the body{\textquoteright}s demands. For example, the human heart continuously adapts in size and geometry to meet greater blood flow needs of the growing body during normal development. In this case, a gradually imposed volume overload leads to progressive chamber enlargement. Another example of a normal physiological growth can be found in the athlete{\textquoteright}s heart, where a sustained elevated chamber pressure results in the chamber wall thickening and an overall increase in cardiac mass. Growth processes, however, can also be maladaptive as found in many cardiovascular diseases where structural changes in the heart progressively decompensate cardiac function.In order to better understand this balance between adaptive and maladaptive cardiac growth, we examine the effect of known growth stimuli using a mechanical model of the heart. We perform a sensitivity analysis of existing growth models in order to assess the relative importance of model parameters and respective mechanisms. This work can eventually lead to simplifications in the model systems for prediction of growth, or help in localizing shortcomings that need to be addressed in the existing modeling frameworks.Methods In order to simulate the motion of the heart throughout the cardiac cycle, we use a nonlinear finite element (FE) model of a realistic left ventricle (LV) coupled to a lumped-parameter model of the systemic circulation. Under the quasi-static assumption, this problem is reduced to finding the displacement u, hydrostatic pressure p and LV pressure p LV that minimize the incompressible strain energy functional Π parameterised by the LV volume V LV [1]: The muscular tissue of the heart is modeled as a transversely isotropic hyperelastic material via the strain energy density Ψ [2]:where m = (a, b, a f , b f ) is a set of passive material parameters, C e is the elastic right Cauchy-Green. By introducing an activation parameter γ representing the active tensor, I1 = trC e and I shortening in the fiber direction f 0 at zero-load, the model incorporates muscle contraction using the active strain approach, which is based on a multiplicative decomposision of the deformation gradient F = I + Grad(u) into an elastic and an active parts F = F e F a with F a defined as:The dynamic changes in the ventricular blood pressure and volume over the entire cardiac cycle are modeled by a three-element Windkessel model described by a system of ordinary differential equations [3]. At each time step, the coupling between the FE model and the circulatory model is achieved through an additional Lagrange multiplier p LV which represents the LV cavity pressure. The problem (1) is solved such that the simulated LV cavity volume V (u) matches the target volume value V LV obtained from the circulatory model.Growth process in the heart wall is modeled by deforming the reference unloaded geometry to a new grown configuration, again through a multiplicative decomposition of F into an elastic and, this time, a growth part, where F = F e F g . The constitutive laws for finite growth can be expressed using a generic format for the growth tensor F g = θ f f 0 (x) f 0 + θ s s 0 (x) s 0 + θ n n 0 (x) n 0. The evolution of the local tissue growth parameter θ g = [θ f , θ s , θ n ] T can be defined in terms of a growth activation function φ (F e ) and a growth rate function k(θ θ g ) which specifies the speed and nonlinearity of the growth process [4].Using the above described model it is possible to simulate various physiological conditions together with the associated structural adaptation of the heart walls in response to change in loadings. In each case, a growth model can be chosen depending on the nature of the considered physiology. The sensitivity of the system{\textquoteright}s grown state to the prescribed growth model can then be estimated. For this, we define an objective functional J(u), the model output of interest, which is to serve as a qualitative and/or quantitative measure of how well the growth model reproduces the expected behavior. More specifically, if we are to compare the model response to a real measurement, then the objective functional can be defined as the mismatch between the simulated u and the measured u exp grown states at a reference time tref :Finally, the sensitivities of J to θ , where θ is a set of growth parameters specific to a given model, are defined as dJ(u)dθ.The solver has been developed within the FEniCS [5] framework and the functional gradients are computed by solving an automatically derived adjoint equations [6].Results We first focus on implementing and testing a strain-driven growth law to simulate a volume overload state of the left ventricle. To achieve this, the initial physiological equilibrium of the heart is altered by increasing a diastolic filling pressure. This initiates cardiac growth that continues until a new equilibrium state is reached. The model is able to reproduce qualitatively experimental observations reported in the literature, such as LV cavity dilation due to fiber over-stretching and a gradual increase in myocardium volume. At the next step, we perform a sensitivity analysis of the model with respect to the growth model parameters and evaluate its performance in reproducing the expected growth behavior.},
url = {http://easychair.org/smart-program/FEniCS{\textquoteright}17/2017-06-12.html$\#$talk:45748},
author = {Nikitushkina, Liubov and Simon W {Funke} and Finsberg, Henrik and Lik Chuan {Lee} and Wall, Samuel}
}
@inproceedings {25451,
title = {Shape Optimization with Multiple Meshes},
journal = {FEniCS Conference 2017},
year = {2017},
address = {University of Luxembourg, Luxembourg},
doi = {10.6084/m9.figshare.5086369},
author = {J{\o}rgen Schartum {Dokken} and Simon W {Funke} and Johansson, August and Schmidt, Stephan}
}
@misc {25420,
title = {Shape Optimization with Multiple Meshes},
howpublished = {FEniCS 2017 conference, Luxembourg},
year = {2017},
abstract = {For shape optimization problems, the computational domain is the design variable. Changing the shape of an airfoil in a channel to minimize drag is such a problem. The evolving domains complicate the numerical solution of shape optimization problems, and typically require large mesh deformations with quality checks and a re-meshing software as a fallback. We propose an approach for solving shape optimization problems on multiple overlapping meshes. In this approach, each mesh can be moved freely and hence the multi-mesh approach allows larger deformation of the domain than standard single-mesh approaches. The approach has been implemented in FEniCS and dolfin-adjoint, by employing the already tested environment for multi-mesh. We give examples of derivation of the shape-optimization problem for a Stokes flow and present implementation of this in FEniCS.Consider a general PDE constrained shape optimization problem we want to minimize a goal functional J, which is subject to a state equation F, with solution u over the domain Omega. We choose to divide the domain Omega into two non-overlapping domains by creating an artificial interface Gamma, such that the union of Omega0 and Omega1 is the original domain Omega. This is depicted in Figure 1. Extension to an arbitrary number of overlapping domains is possible. The weak formulation of the state equations are then formulated and the continuity over the artificial boundary is enforced by using Nitsches method.For minimization, we choose a gradient based scheme, and find the gradient by using the adjoint method. By employing the Hadamard formulas for Volume and Surface objective functions one can achieve the functional sensitivities as a function of the moving boundary and not the domain.A concrete example of this approach is the shape optimization of an obstacle in Stokes-flow in the domain specified in Figure 2.For deformation of the mesh, we have used two different deformation equations, a Laplacian smoothing and a set of Eikonal convection equations. For the multi-mesh problem, deformation is only done one the front mesh, while the background mesh is stationary. Figure 3 shows that with the Laplacian deformation the mesh degenerates in both the single-mesh and multi-mesh-case. Figure 4 shows that the Eikonal convection equations preserves the mesh-quality in the multi-mesh-case, but not in the single-mesh case, where the mesh degenerates at the boundary. We conclude that with a multi-mesh-approach, the meshes are preserved better than with a single-mesh approach.},
url = {http://easychair.org/smart-program/FEniCS{\textquoteright}17/2017-06-13.html},
author = {J{\o}rgen Schartum {Dokken} and Simon W {Funke} and Johansson, August and Schmidt, Stephan}
}
@misc {25590,
title = {Shape Optimization with Overlapping Meshes},
howpublished = {ENUMATH 2017, Voss, Bergen},
year = {2017},
type = {Invited speaker to Minisymposia on PDE Software Frameworks},
keywords = {Adjoint Method, CUTFEM, MultiMesh, Shape-optimization},
author = {J{\o}rgen Schartum {Dokken}},
editor = {Simon W {Funke} and Johansson, August and Schmidt, Stephan}
}
@article {25610,
title = {A surrogate-model assisted approach for optimising the size of tidal turbine arrays},
journal = {International Journal of Marine Energy},
volume = {19},
number = {Supplement C},
year = {2017},
pages = {357 - 373},
publisher = {Elsevier},
keywords = {Array layout},
issn = {2214-1669},
doi = {https://doi.org/10.1016/j.ijome.2017.05.001},
url = {http://www.sciencedirect.com/science/article/pii/S2214166917300462},
author = {Culley, D.M. and Simon W {Funke} and Kramer, S.C. and Piggott, M.D.}
}
@article {25603,
title = {The trade-off between tidal-turbine array yield and impact on flow: A multi-objective optimisation problem},
journal = {Renewable Energy},
volume = {114},
number = {Part B},
year = {2017},
pages = {1247 - 1257},
publisher = {Elsevier},
keywords = {Environmental impact},
issn = {0960-1481},
doi = {https://doi.org/10.1016/j.renene.2017.07.081},
url = {http://www.sciencedirect.com/science/article/pii/S0960148117307097},
author = {R.J. du {Feu} and Simon W {Funke} and Kramer, S.C. and Culley, D.M. and Hill, J. and B S {Halpern} and Piggott, M.D.}
}
@misc {24877,
title = {Apparatur and Method for Global Optimization},
year = {2016},
author = {Noack, Marcus and Simon W {Funke}}
}
@article {23601,
title = {Design optimisation and resource assessment for tidal-stream renewable energy farms using a new continuous turbine approach},
journal = {Renewable Energy},
volume = {99},
year = {2016},
pages = {1046-1061},
publisher = {Elsevier},
abstract = {This paper presents a new approach for optimising the design of tidal stream turbine farms. In this approach, the turbine farm is represented by a turbine density function that specifies the number of turbines per unit area and an associated continuous locally-enhanced bottom friction field. The farm design question is formulated as a mathematical optimisation problem constrained by the shallow water equations and solved with efficient, gradient-based optimisation methods. The resulting method is accurate, computationally efficient, allows complex installation constraints, and supports different goal quantities such as to maximise power or profit. The outputs of the optimisation are the optimal number of turbines, their location within the farm, the overall farm profit, the farm\&$\#$39;s power extraction, and the installation cost. We demonstrate the capabilities of the method on a validated numerical model of the Pentland Firth, Scotland. We optimise the design of four tidal farms simultaneously, as well as individually, and study how farms in close proximity may impact upon one another.},
doi = {doi:10.1016/j.renene.2016.07.039},
url = {http://www.sciencedirect.com/science/article/pii/S0960148116306358},
author = {Simon W {Funke} and Stephan C. {Kramer} and Matthew David {Piggott}}
}
@misc {24566,
title = {Designing Tidal Turbine Arrays With PDE-constrained Optimisation},
howpublished = {Pilsen, Czech Republic},
year = {2016},
author = {Simon W {Funke}}
}
@misc {25076,
title = {Designing Tidal Turbine Arrays With PDE-constrained Optimisation},
howpublished = {ESCO 2016, Plze{\v n}, Czech Republic},
year = {2016},
author = {Simon W {Funke}}
}
@article {23605,
title = {Integration of cost modelling within the micro-siting design optimisation of tidal turbine arrays},
journal = {Renewable Energy},
volume = {85},
year = {2016},
month = {01/2016},
pages = {215-227},
publisher = {Elsevier},
abstract = {Abstract The location of individual turbines within a tidal current turbine array {\^a}\€\“ micro-siting {\^a}\€\“ can have a significant impact on the power that the array may extract from the flow. Due to the infancy of the industry and the challenges of exploiting the resource, the economic costs of realising industrial scale tidal current energy projects are significant and should be considered as one of the key drivers of array design. This paper proposes a framework for the automated design of tidal current turbine arrays in which costs over the lifespan of the array may be modelled and considered as part of the design optimisation process. To demonstrate this approach, the cost of sub-sea cabling is incorporated by implementing a cable-routing algorithm alongside an existing gradient-based array optimisation algorithm. Three idealised test scenarios are used to demonstrate the effects of a financial-return optimising design approach as contrasted with a power maximisation approach.},
keywords = {Financial-return},
doi = {http://dx.doi.org/10.1016/j.renene.2015.06.013},
url = {http://www.sciencedirect.com/science/article/pii/S0960148115300379},
author = {Culley, D.M. and Simon W {Funke} and Kramer, S.C. and Matthew David {Piggott}}
}
@misc {25075,
title = {Introduction to FEniCS and dolfin-adjoint},
howpublished = {Symposium on the Application of Finite Elements in Physics and Engineering, University of South Africa, Johannesburg, South Africa},
year = {2016},
author = {Simon W {Funke}}
}
@misc {24900,
title = {Optimization Problems on Dynamical Domains with Non-Matching Meshes},
howpublished = {29th Nordic Seminar on Computational mechanics},
year = {2016},
keywords = {Adjoint, Dynamic Domains, Finite element method, Nitsche, Non-matching meshes, Optimization},
author = {J{\o}rgen Schartum {Dokken}},
editor = {Johansson, August and Simon W {Funke}}
}
@inproceedings {24197,
title = {On the validity of tidal turbine array configurations obtained from steady-state adjoint optimisation},
journal = {European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016)},
volume = {4},
year = {2016},
month = {09/2016},
pages = {8247-8261},
publisher = {Institute of Structural Analysis and Antiseismic Research, National Technical University of Athens, Greence},
abstract = {Extracting the optimal amount of power from an array of tidal turbines requires an intricate understanding of tidal dynamics and the effects of turbine placement on the local and regional scale flow. Numerical models have contributed significantly towards this understand- ing, and more recently, adjoint-based modelling has been employed to optimise the positioning of the turbines in an array in an automated way and improve on simple man-made configurations (e.g. structured grids of turbines) [15]. Adjoint-based optimisation of high-resolution and ideally 3D transient models is generally a very computationally expensive problem. Multiple approaches are therefore used in practice to obtain feasible runtimes: using high viscosity values to obtain a steady-state solution, or a sequence of steady-state solutions for \“time-varying\” setups; limiting the number of adjoint computations; or reformulating the problem to allow for coarser mesh resolution to make it feasible for resources assessment (e.g. [16], [4]). However, such compromises may affect the reliability of the modelled turbines, their wakes and interactions, and thus bring into question the validity of the computed optimal turbine positions. This work considers a suite of idealised simulations of flow past tidal turbine arrays in a two-dimensional channel. It compares four regular array configurations, detailed by Divett et al. [8], with the configuration found through adjoint optimisation in a steady-state, high- viscosity setup. The optimised configuration produces considerably more power than the other configurations (approximately 40\% more than the best man-made configuration). The same configurations are then used to produce a suite of transient simulations that do not use con- stant high-viscosity, and instead use large-eddy simulation (LES) to parameterise the resulting turbulent structures. All simulations are performed using OpenTidalFarm [15]. It is shown that the \‘low background viscosity\’/LES simulations produce less power than that predicted by the constant high-viscosity runs. Nevertheless, they still follow the same trends in the power curve throughout time, with optimised layouts continuing to perform significantly better than simplified configurations.},
isbn = {978-618-82844-0-1},
url = {https://drive.google.com/open?id=0B04URlVl3wyuMmRLLW9GMkZiN0E},
author = {Christian T. {Jacobs} and Matthew David {Piggott} and Stephan C. {Kramer} and Simon W {Funke}},
editor = {Papadrakakis, M. and Papadopoulos, V. and Stefanou, G. and Plevris, V.}
}
@misc {24815,
title = {Variational data assimilation for blood flow simulations},
howpublished = { MS at SIAM UQ, Zurich},
year = {2016},
author = {Mardal, Kent-Andre and Simon W {Funke} and Nordaas, Magne and Evju, {\O}yvind and Martin Sandve {Aln{\ae}s}}
}
@misc {24605,
title = {Assimilating 4D-MRI blood flow measurements using PDE-constrained optimisation},
howpublished = {Oslo},
year = {2015},
publisher = {Workshop and Advanced Numerical Techniques in Biomedical Computing},
author = {Simon W {Funke}}
}
@inproceedings {23603,
title = {A continuous approach for the optimisation of tidal turbine farms},
journal = {European Wave and Tidal Energy Conference 2015},
year = {2015},
abstract = {The optimal placement of individual devices within a tidal turbine farm is \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a challenging problem that may benefit greatly from automated optimisation methods. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A previously published gradient-based approach for the optimal placement/arrangement \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ of turbines significantly reduces the number of required iterations \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ over traditional optimisation methods. This allows for a higher level of computational cost, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ and hence realism, in the hydrodynamic model used in every optimisation iteration. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Here, we \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ introduce a closely related approach that optimises for a turbine density field \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (referred to here as the {\textquoteleft}{\textquoteleft}continuous{\textquoteright}{\textquoteright} approach) instead of the positions of individual turbines \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (the {\textquoteleft}{\textquoteleft}discrete{\textquoteright}{\textquoteright} approach). \ Its advantages are: (1) it requires less \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mesh resolution than the discrete approach and hence has lower computational costs; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) the number of turbines does not need to be chosen in advance -- this allows for \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ the inclusion of per-turbine-costs to be included in the optimisation, and as a by-product \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ returns an estimate for the optimal number of turbines on a site; (3) it allows for the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ inclusion of more complex site design constraints. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ We present a number of cases to demonstrate the validity of the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ method. \ The optimal number of turbines predicted by the continuous approach is \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ shown to agree well with the results of running several discrete optimisations \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ with different numbers of turbines, giving confidence to the validity of the new \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ approach. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ In a realistic case we show how non-convex domain sites can be optimised. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Furthermore, this approach naturally supports complex constraints such as \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ maximum bathymetry gradients above which turbines cannot be installed, and \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ the simultaneous optimisation of multiple, potentially interacting farms.\ },
author = {Stephan C {Kramer} and Simon W {Funke} and Matthew David {Piggott}}
}
@misc {23859,
title = {Data assimilation in time-dependent blood flow simulations},
howpublished = {Ankara},
year = {2015},
publisher = {ENUMATH 2015},
type = {Oral},
author = {Simon W {Funke}}
}
@misc {23861,
title = {Data assimilation in time-dependent blood flow simulations},
howpublished = {London, UK},
year = {2015},
publisher = {FEniCS {\textquoteright}15 Conference},
type = {Oral},
author = {Simon W {Funke}}
}
@article {Simula.simula.2978,
title = {Deflation Techniques for Finding Distinct Solutions of Nonlinear Partial Differential Equations},
journal = {SIAM Journal on Scientific Computing},
volume = {37},
year = {2015},
pages = {A2026-A2045},
publisher = {SIAM},
abstract = {Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton{\textquoteright}s method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton-Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations are observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and fluid mechanics that permit distinct solutions.},
doi = {10.1137/140984798},
author = {Patrick Emmet {Farrell} and Birkisson, Asgeir and Simon W {Funke}}
}
@article {Simula.simula.2944,
title = {Designing Large Arrays of Tidal Turbines: a Synthesis and Review},
journal = {Renewable \& Sustainable Energy Reviews},
volume = {41},
year = {2015},
pages = {454-472},
publisher = {Elsevier},
abstract = {Much of the global tidal current energy resource lies in the accelerated flows along narrow channels. These channels have the potential to produce 10s -1000s of MW of electricity. However, realizing 100 MW of a channel{\textquoteright}s potential is much more complex than installing 100 one MW turbines because large scale power extraction reduces tidal currents throughout the channel, changing the resource. This synthesis and review gives an overview of the issues and compromises in designing the layout of the large tidal turbine arrays required to realize this potential. The paper focuses on macro- and micro-design of arrays. Macro-design relates to the total number of turbines and their gross arrangement into rows, while micro-design adjusts the relative positions of the turbines within a grid and the spacing between rows. Interdependent macro-design compromises balance the total number of turbines, array power output, the power output of each turbine, the loads turbines experience, turbine construction costs, maintaining navigability along the channel and any environmental impacts due to flow reduction. A strong emphasis is placed on providing physical insights about how {\textquotedblleft}channel-scale dynamics{\textquotedblright} and the {\textquotedblleft}duct-effect{\textquotedblright} impact on the compromises in ar- ray design. This work is relevant to the design of any {\textquotedblleft}large{\textquotedblright} array which blocks more than 2\%-5\% of a channel{\textquoteright}s cross-section, be it 2 turbines in a small channel or 100 turbines in a large channel.},
doi = {10.1016/j.rser.2014.08.022},
author = {Venell, Ross and Simon W {Funke} and Draper, Scott and Stevens, Craig and Divett, Tim}
}
@article {23712,
title = {Determining Tidal Turbine Farm Efficiency in the Western Passage using the Disc Actuator Theory},
journal = {Ocean Dynamics},
volume = {66},
year = {2015},
pages = {41-57},
publisher = {Springer},
abstract = {Tidal power potential is determined across the Western Passage in Passamaquoddy Bay using the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Finite Volume Community Ocean Model (FVCOM). The tidal turbines are implemented in FVCOM \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ using the disc actuator theory method to determine the power potential for different densities and \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ arrangements of tidal turbines. At the most efficient setting for 10 turbines across the Western Pas- \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sage, the optimal turbine drag coefficient is 2.0 and the average power output, in a 2-week period, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ is \∼819 kW. Results suggest that for a single row of turbines, the addition of turbines decreases the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ efficiency of the turbine farm, but this decrease in efficiency is less than 7\%. A parallel distribution of \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ turbines in an array diminishes the average power for turbines in the shadow of other turbines, while \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ staggered distribution in an array increases the average power extraction for some turbines, due to the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ speed gains in the gaps between turbines. A simple tidal farm optimization using the OpenTidalFarm \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (OTF) model suggests a similar tidal farm distribution.},
keywords = {Bay of Fundy, Disc actuator theory, Farm optimization, Tidal power, Tidal turbine efficiency, Tidal turbine farms, Western passage},
issn = {1616-7341},
doi = {10.1007/s10236-015-0906-y},
url = {http://dx.doi.org/10.1007/s10236-015-0906-y},
author = {Rao, Shivanesh and Xue, Huijie and Bao, Min and Simon W {Funke}}
}
@misc {24607,
title = {Dolfin-adjoint, automated adjoint models for FEniCS},
year = {2015},
note = {Winner of the Best Poster Award},
publisher = {SIAM Conference on Computational Science and Engineering},
address = {Salt Lake City, USA},
author = {Simon W {Funke} and Patrick Emmet {Farrell} and Ham, D.A. and Marie E. {Rognes}}
}
@misc {23582,
title = {Dolfin-adjoint: Automatic adjoint models for FEniCS},
howpublished = {The 8th International Congress on Industrial and Applied Mathematics},
year = {2015},
publisher = {The 8th International Congress on Industrial and Applied Mathematics},
address = {Beijing, China},
abstract = {Adjoint and tangent linear models form the basis of many numerical techniques, including sensitivityanalysis, optimization and stability analysis. The implementation of adjoint models for nonlinear or time-dependent models are notoriously challenging: the manual approach is time-consuming and traditionalautomatic differentiation tools lack robustness and performance.dolfin-adjoint solves this problem by automatically analyzing the high-level mathematical structureinherent in finite element methods. It raises the traditional abstraction of algorithmic differentiationfrom the level of individual floating point operations to that of whole systems of differential equations.This approach delivers a number of advantages over the previous state-of-the-art: robust hands-offautomation of adjoint model derivation, optimal computational efficiency, and native parallel support.},
author = {Simon W {Funke} and Patrick Emmet {Farrell} and David A. {Ham} and Marie E. {Rognes}}
}
@misc {24606,
title = {Mesh-Independent Convergence for PDE-Constrained Optimisation Solvers in Dolfin-Adjoint},
howpublished = {Salt Lake City, USA},
year = {2015},
publisher = {SIAM Conference on Computational Science and Engineering},
type = {Oral},
author = {Simon W {Funke}}
}
@misc {24604,
title = {Optimizing tidal turbine farms with high-level tools},
howpublished = {Texas A\&M, USA},
year = {2015},
publisher = {Special Oceanography Seminar},
type = {Oral},
address = {College Station, USA},
author = {Simon W {Funke} and Roc, Thomas}
}
@inproceedings {23604,
title = {Standard methodology for tidal array project optimisation: An idealized study of the Minas Passage},
journal = {European Wave and Tidal Energy Conference 2015},
year = {2015},
abstract = {This proceeding describes the principles as well as an idealised application of a standard methodology designed for tidal array project optimisation. In essence, this iterative method defines, in a systematic fashion, how to design the most adapted optimisation strategy to a particular project. This coupled optimisation system accounts for both hydrodynamic and techno-economic aspects and has lead to an improvement of 63\% of the internal rate of return compared to a non-optimised scenario. This proceeding also investigates comprehensive metrics and benchmarks for tidal array optimisation based on both environmental and socio-economic aspects of the project site by investigating the sensitivity of a set of optimisation levers and analysing array-induced hydrodynamic-impacts. Future work will involve accounting for additional constraints based on limiting environmental impacts.\ },
author = {Roc, Thomas and Simon W {Funke} and Kristen M. {Thyng}}
}
@inproceedings {23602,
title = {Tidal stream resource assessment through optimisation of array design with quantification of uncertainty},
journal = { European Wave and Tidal Energy Conference},
year = {2015},
abstract = {As the number of tidal stream turbines within an array increases, so the effects of \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ that array on the flow, environment, and on the objective measures (such as \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ energy yield or financial profit) begin to suffer the effects of \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ diminishing increases. In addition, it has been demonstrated that \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ optimising the layout of individual \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ turbines within an array -- micro-siting -- facilitates significantly increased energy \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ harvest for a given number of turbines in a given area, as compared to a {\textquoteleft}human{\textquoteright}s \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ best attempt{\textquoteright} design. The optimal micro-siting design of an array, and therefore \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ an accurate forecast of the yield of that array, must be found as the product of an \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ optimisation exercise which may incorporate turbine parameters, local bathymetry and \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a host of other practical, physical, legal, financial or environmental constraints. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ The number of turbines within an array proposed for a given site has \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ an even greater effect on the yield of that array and is consequently a problem \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ of critical importance. Like the micro-siting design, this also cannot be \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ solved directly but must be approached through a process of iterative \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ optimisation. Consequently, only through determining the optimal number of turbines \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ and their arrangement, can a reliable estimate of the accessible tidal \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ resource on a site be made. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Having determined the optimal array design it is imperative that the sensitivity \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ of the design to the modelling assumptions can be quantified and fully understood. In this paper \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a framework for the design of tidal turbine arrays (both size and micro-siting) is presented \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ with functionality to measure the sensitivity of the objective function with respect to the \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ modelling parameters. \ \ },
author = {Dave M {Culley} and Simon W {Funke} and Stephan C {Kramer} and Matthew David {Piggott}}
}
@misc {Simula.simula.3126,
title = {Benchmarking Tidal Array Optimization: a Balance Between Impacts \& Economics of the Bay of Fundy},
howpublished = {EWTEC 2015},
year = {2014},
month = {November},
abstract = {To date, some 190 tidal power sites have been identified off Canada\&$\#$39;s coasts, with a total estimated capacity of 42,000 MW equivalent to more than 63 \% of the country\&$\#$39;s annual total consumption. Tidal Energy Converters (TECs) deployed in arrays appear to be the most promising solution to efficiently capture this carbon neutral energy resource. Indeed, in contrast to tidal bar-rages, TEC arrays permit a higher concentration of devices without entirely blocking the tidal flow and thus avoid drastic changes in the hydrodynamics of the site while minimizing the overall cost,by allowing for shared maintenance and grid connection expenses. Whilst the tidal energy industry grows and commercial-scale TEC array projects emerge, the optimisation of TEC array becomes prudent to improve financial viability and limit environmental impacts. The very concept of TEC array optimisation covers a multitude of scopes depending on one\&$\#$39;s goals and interests. The set of levers, benchmarks and constraints would vary from the point of view of a technology developer, a project developer, a site manager, an investor, a policy maker and so on. Consequently different sets of Environmental, technical and economic parameters shall be used for different optimisation scopes.Unfortunately, being the most efficient solution does not mean it is the simplest one. As a matter of fact, TEC array optimisation may encompass numerous parameters, all inter-connected through the hydrodynamics governing the flow surrounding the considered array. Such intricate problem requires cutting-edge numerical methods in order to be appropriately addressed. A collaboration between the authors has enabled this study to move one step forward by numerically simulating both their potential hydrodynamic impacts and financial viability to help understand the leveraging importance of each of these aspects as well as potential brakes on project development, ultimately lowering the risks of project failure and thus stimulating investment.},
keywords = {Conference},
author = {Roc, Thomas and Thyng, Kristen and Simon W {Funke}}
}
@article {dolfin-adjoint-gst,
title = {A Framework for the Automation of Generalised Stability Theory},
journal = {SIAM Journal on Scientific Computing},
volume = {36},
year = {2014},
month = {01/2014},
pages = {C25{\textendash}C48},
publisher = {SIAM},
abstract = {The traditional approach to investigating the stability of a physical system is to linearize the equations about a steady base solution, and to examine the eigenvalues of the linearized operator. Over the past several decades, it has been recognized that this approach only determines the asymptotic stability of the system, and neglects the possibility of transient perturbation growth arising due to the nonnormality of the system. This observation motivated the development of a more powerful generalized stability theory (GST), which focuses instead on the singular value decomposition (SVD) of the linearized propagator of the system. While GST has had significant successes in understanding the stability of phenomena in geophysical fluid dynamics, its more widespread applicability has been hampered by the fact that computing the SVD requires both the tangent linear operator and its adjoint: deriving the tangent linear and adjoint models is usually a considerable challenge, and manually embedding them inside an eigensolver is laborious. In this paper, we present a framework for the automation of generalized stability theory, which overcomes these difficulties. Given a compact high-level symbolic representation of a finite element discretization implemented in the FEniCS system, efficient C++ code is automatically generated to assemble the forward, tangent linear, and adjoint models; these models are then used to calculate the optimally growing perturbations to the forward model, as well as their growth rates. By automating the stability computations, we hope to make these powerful tools a more routine part of computational analysis. The efficiency and generality of the framework are demonstrated, with applications drawn from geophysical fluid dynamics, phase separation, and quantum mechanics.\ },
doi = {10.1137/120900745},
author = {Patrick Emmet {Farrell} and Cotter, Colin and Simon W {Funke}}
}
@inproceedings {Simula.simula.3080,
title = {A Hierarchy of Approaches for the Optimal Design of Tidal Turbine Arrays},
journal = {ICOE 2014 (5th International Conference on Ocean Energy)},
year = {2014},
type = {1},
abstract = {From conception to construction, the process by which tidal turbine farms are scoped and designed (and even optimised - which is the focus here) is multi-layered. The industrial designer requires tools of varying fidelities working on multiple scales, depending on the task at hand. In this paper a hierarchy of modelling approaches is proposed and some examples demonstrated. For site-scoping and resource assessment, the continuum approach enables multiple farms to be considered and optimised over a large geographic area. This is demonstrated for four farms in the Pentland Firth, Scotland. For detailed design, three-dimensional CFD codes allow flow around a turbine to be fully resolved and the physical processes closely modelled. In between, and informed by these extremes, are array design tools whereby each turbine is individually represented and the flow over the domain is calculated with the non-linear shallow water equations. In a test example, the positions of 78 turbines in a farm located in the Inner Sound of the Pentland Firth, Scotland is optimised with a resulting 25\% improvement in power extracted. A holistic approach to the design process is also presented which seeks to design with the maximisation of developer\&$\#$39;s profit - rather than power extracted - as the ultimate goal.},
author = {Dave M. {Culley} and Simon W {Funke} and S. C. {Kramer} and Matthew David {Piggott}}
}
@article {Simula.simula.2946,
title = {Hybrid Global-Local Optimisation Algorithms for the Layout Design of Tidal Turbine Arrays},
journal = {Renewable Energy},
year = {2014},
publisher = {Elsevier},
abstract = {Tidal stream power generation represents a promising source of renewable energy. In order to extract an economically useful amount of power, tens to hundreds of tidal turbines need to be placed within an array. The layout of these turbines can have a significant impact on the power extracted and hence on the viability of the site. Funke et al. [13] formulated the question of the best turbine layout as an optimisation problem constrained by the shallow water equations and solved it using a local, gradient-based optimisation algorithm. Given the local nature of this approach, the question arises of how optimal the layouts actually are. This becomes particularly important for scenarios with complex bathymetry and layout constraints, both of which typically introduce locally optimal layouts. Optimisation algorithms which find the global optima generally require orders of magnitude more iterations than local optimisation algorithms and are thus infeasible in combination with an expensive flow model. This paper presents an analytical wake model to act as an efficient proxy to the shallow water model. Based upon this, a hybrid global-local two-stage optimisation approach is presented in which turbine layouts are first optimised with the analytical wake model via a global optimisation algorithm, and then further optimised with the shallow water model via a local gradient-based optimisation algorithm. This procedure is applied to a number of idealised cases and a more realistic case with complex bathymetry in the Inner Sound of the Pentland Firth, Scotland. It is shown that in cases where bathymetry is considered, the two-stage optimisation procedure is able to improve the power extracted from the array by as much as 25 \% compared to local optimisation for idealised scenarios and by as much as 12 \% for the more realistic Pentland Firth scenario whilst in many cases reducing the overall computation time by approximately 30 \% to 40 \%.},
author = {George L. {Barnett} and Simon W {Funke} and Matthew David {Piggott}}
}
@misc {23455,
title = {Lecture on PDE-constrained optimization with FEniCS},
howpublished = {Zhjiang University, Hangzhou, China},
year = {2014},
type = {Invited},
author = {Simon W {Funke}}
}
@misc {Simula.simula.2950,
title = {PDE-Constrained Optimisation in Hilbert Spaces},
howpublished = {FEniCS{\textquoteright}14 workshop, Paris},
year = {2014},
month = {June},
abstract = {The solution of optimisation problems constrained by partial differential equations becomes increasingly more feasible due to the increase of computational resources and the development of solution algorithms. The computational cost and the large number of degrees of freedoms in PDE applications raise a particular challenge for partical algorithms. A key property for an efficient optimisation algorithm is that the required number of iterations is independent on the local mesh refinement and its element size. However, a straight-forward application on standard optimisation algorithms on the reduced optimisation problem results in mesh-size dependent iteration numbers and a poor performance on non-homogoenous meshes. In this talk we present on how good convergence is obtained by rethinking these optimisation algorithms to respect the underlying inner products and induced norms of the involved function spaces. Based on this idea we develop a new optimisation framework specifically designed for PDE-constrained optimisation which honors the inner product of the underlying Hilbert spaces. We show that this strategy is equivalent to a variance of preconditioned conjugate gradient methods or for the BFGS a user defined initial estimate of the Hessian matrix. Parallel , currently supports unconstrained optimisation algorithms and integrates seemlesly with FEniCS and dolfin-adjoint. Numerical results demonstrate the effectiveness of the framework.},
keywords = {Conference},
author = {Simon W {Funke} and Nordaas, Magne}
}
@misc {Simula.simula.3128,
title = {Resource Assessment of Tidal Sites Through Array Optimisation of the Number of Turbines and the Micro-Siting Design},
year = {2014},
month = {November},
abstract = {\~ \~},
keywords = {Conference},
author = {Dave M. {Culley} and Simon W {Funke} and Kramer, Stephan and Matthew David {Piggott}}
}
@article {Simula.simula.2248,
title = {Tidal Turbine Array Optimisation Using the Adjoint Approach},
journal = {Renewable Energy},
volume = {63},
year = {2014},
month = {March},
pages = {658-673},
publisher = {Elsevier},
abstract = {Oceanic tides have the potential to yield a vast amount of renewable energy. Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of power, hundreds of tidal turbines must typically be deployed in an array. This naturally leads to the question of how these turbines should be configured to extract the maximum possible power: the positioning and the individual tuning of the turbines could significantly influence the extracted power, and hence is of major economic interest. However, manual optimisation is difficult due to legal site constraints, nonlinear interactions of the turbine wakes, and the cubic dependence of the power on the flow speed. The novel contribution of this paper is the formulation of this problem as an optimisation problem constrained by a physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element shallow water model predicts the flow and the performance of the current array configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of the time taken for a flow solution by solving the associated adjoint equations. These equations propagate causality backwards through the computation, from the power extracted back to the turbine positions and the tuning parameters. This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach is demonstrated by optimising turbine arrays in four idealised scenarios and a more realistic case with up to 256 turbines in the Inner Sound of the Pentland Firth, Scotland.},
doi = {10.1016/j.renene.2013.09.031},
author = {Simon W {Funke} and Patrick Emmet {Farrell} and Matthew David {Piggott}}
}
@article {dolfinadjoint,
title = {Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs},
journal = {SIAM Journal on Scientific Computing},
volume = {35},
number = {4},
year = {2013},
pages = {369-393},
author = {Patrick Emmet {Farrell} and David A. {Ham} and Simon W {Funke} and Marie E. {Rognes}}
}
@article {Funke20111483,
title = {A wetting and drying algorithm with a combined pressure/free-surface formulation for non-hydrostatic models},
journal = {Advances in Water Resources},
volume = {34},
year = {2011},
pages = {1483 - 1495},
publisher = {Elsevier},
abstract = {A wetting and drying method for free-surface problems for the three-dimensional, non-hydrostatic Navier{\^a}{\texteuro}{\textquotedblleft}Stokes equations is proposed. The key idea is to use a horizontally fixed mesh and to apply different boundary conditions on the free-surface in wet and dry zones. In wet areas a combined pressure/free-surface kinematic boundary condition is applied, while in dry areas a positive water level and a no-normal flow boundary condition are enforced. In addition, vertical mesh movement is performed to accurately represent the free-surface motion. Non-physical flow in the remaining thin layer in dry areas is naturally prevented if a Manning{\^a}{\texteuro}{\textquotedblleft}Strickler bottom drag is used. The treatment of the wetting and drying processes applied through the boundary condition yields great flexibility to the discretisation used. Specifically, a fully unstructured mesh with any finite element choice and implicit time discretisation method can be applied. The resulting method is mass conservative, stable and accurate. It is implemented within Fluidity-ICOM [1] and verified against several idealized test cases and a laboratory experiment of the Okushiri tsunami.},
doi = {http://dx.doi.org/10.1016/j.advwatres.2011.08.007},
url = {http://www.sciencedirect.com/science/article/pii/S0309170811001564},
author = {Simon W {Funke} and Pain, C.C. and Kramer, S.C. and Matthew David {Piggott}}
}