Imperial College London

Joe Wallwork

Faculty of EngineeringDepartment of Earth Science & Engineering

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Royal School of MinesSouth Kensington Campus

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Publications

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10 results found

Wallwork JG, Angeloudis A, Barral N, Mackie L, Kramer SC, Piggott MDet al., 2024, Tidal turbine array modelling using goal-oriented mesh adaptation, Journal of Ocean Engineering and Marine Energy, Vol: 10, Pages: 193-216, ISSN: 2198-6452

To examine the accuracy and sensitivity of tidal array performance assessment by numerical techniques applying goal-oriented mesh adaptation. The goal-oriented framework is designed to give rise to adaptive meshes upon which a given diagnostic quantity of interest (QoI) can be accurately captured, whilst maintaining a low overall computational cost. We seek to improve the accuracy of the discontinuous Galerkin method applied to a depth-averaged shallow water model of a tidal energy farm, where turbines are represented using a drag parametrisation and the energy output is specified as the QoI. Two goal-oriented adaptation strategies are considered, which give rise to meshes with isotropic and anisotropic elements. We present both fixed mesh and goal-oriented adaptive mesh simulations for an established test case involving an idealised tidal turbine array positioned in a channel. With both the fixed meshes and the goal-oriented methodologies, we reproduce results from the literature which demonstrate how a staggered array configuration extracts more energy than an aligned array. We also make detailed qualitative and quantitative comparisons between the fixed mesh and adaptive outputs. The proposed goal-oriented mesh adaptation strategies are validated for the purposes of tidal energy resource assessment. Using only a tenth of the number of degrees of freedom as a high-resolution fixed mesh benchmark and lower overall runtime, they are shown to enable energy output differences smaller than 2% for a tidal array test case with aligned rows of turbines and less than 10% for a staggered array configuration.

Journal article

Kärnä T, Wallwork JG, Kramer SC, 2023, Efficient optimization of a regional water elevation model with an automatically generated adjoint, Journal of Advances in Modeling Earth Systems, Vol: 15, ISSN: 1942-2466

Calibration of unknown model parameters is a common task in many ocean model applications. We present an adjoint-based optimization of an unstructured mesh shallow water model for the Baltic Sea. Spatially varying bottom friction parameter is tuned to minimize the misfit with respect to tide gauge sea surface height (SSH) observations. A key benefit of adjoint-based optimization is that computational cost does not depend on the number of unknown variables. Adjoint models are, however, typically very laborious to implement. In this work, we leverage a domain specific language framework in which the discrete adjoint model can be obtained automatically. The adjoint model is both exactly compatible with the discrete forward model and computationally efficient. A gradient-based quasi-Newton method is used to minimize the misfit. Optimizing spatially-variable parameters is typically an under-determined problem and can lead to over-fitting. We employ Hessian-based regularization to penalize the spatial curvature of the friction field to overcome this problem. The SSH dynamics in the Baltic Sea are simulated for a 3-month period. Optimization of the bottom friction parameter results in significant improvement of the model performance. The results are especially encouraging in the complex Danish Straits region, highlighting the benefit of unstructured meshes. Domain specific language frameworks enable automated model analysis and provide easy access to adjoint modeling. Our application shows that this capability can be enabled with few efforts, and the optimization procedure is robust and computationally efficient.

Journal article

Clare MCA, Wallwork JG, Kramer SC, Weller H, Cotter CJ, Piggott MDet al., 2022, Multi-scale hydro-morphodynamic modelling using mesh movement methods, GEM: International Journal on Geomathematics, Vol: 13, ISSN: 1869-2672

Hydro-morphodynamic modelling is an important tool that can be used in the protection of coastal zones. The models can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement methods are particularly well-suited to coastal problems as they allow the mesh to move in response to evolving flow and morphology structures. This new capability is demonstrated using test cases that exhibit complex evolving bathymetries and have moving wet-dry interfaces. In order to be able to simulate sediment transport in wet-dry domains, a new conservative discretisation approach has been developed as part of this work, as well as a sediment slide mechanism. For all test cases, we demonstrate how mesh movement methods can be used to reduce discretisation error and computational cost. We also show that the optimum parameter choices in the mesh movement monitor functions are fairly predictable based upon the physical characteristics of the test case, facilitating the use of mesh movement methods on further problems.

Journal article

Song W, Wallwork JG, Tian Z, Piggott MD, Zhang M, Gao J, Sun F, Chen J, Shi Z, Chen X, Wang Jet al., 2022, M2N: Mesh movement networks for PDE solvers, 36th Conference on Neural Information Processing Systems (NeurIPS 2022), Pages: 1-12, ISSN: 1049-5258

Numerical Partial Differential Equation (PDE) solvers often require discretizing the physical domain by using a mesh. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without introducing extra computational burden to the PDE solver, by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, sophisticated mesh movement methods, such as the Monge-Ampère method, generally require the solution of auxiliary equations. These solutions can be extremely expensive to compute when the mesh needs to be adapted frequently. In this paper, we propose to the best of our knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are: alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large mesh deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampère method, our approach can greatly accelerate the mesh adaptation process by three to four orders of magnitude, whilst achieving comparable numerical error reduction.

Conference paper

Song W, Zhang M, Wallwork JG, Gao J, Tian Z, Sun F, Piggott MD, Chen J, Shi Z, Chen X, Wang Jet al., 2022, M2N: mesh movement networks for PDE solvers

Mainstream numerical Partial Differential Equation (PDE) solvers requirediscretizing the physical domain using a mesh. Mesh movement methods aim toimprove the accuracy of the numerical solution by increasing mesh resolutionwhere the solution is not well-resolved, whilst reducing unnecessary resolutionelsewhere. However, mesh movement methods, such as the Monge-Ampere method,require the solution of auxiliary equations, which can be extremely expensiveespecially when the mesh is adapted frequently. In this paper, we propose toour best knowledge the first learning-based end-to-end mesh movement frameworkfor PDE solvers. Key requirements of learning-based mesh movement methods arealleviating mesh tangling, boundary consistency, and generalization to meshwith different resolutions. To achieve these goals, we introduce the neuralspline model and the graph attention network (GAT) into our modelsrespectively. While the Neural-Spline based model provides more flexibility forlarge deformation, the GAT based model can handle domains with more complicatedshapes and is better at performing delicate local deformation. We validate ourmethods on stationary and time-dependent, linear and non-linear equations, aswell as regularly and irregularly shaped domains. Compared to the traditionalMonge-Ampere method, our approach can greatly accelerate the mesh adaptationprocess, whilst achieving comparable numerical error reduction.

Working paper

Wallwork JG, Barral N, Ham DA, Piggott MDet al., 2022, Goal-oriented error estimation and mesh adaptation for tracer transport modelling, Computer-Aided Design, Vol: 145, Pages: 1-21, ISSN: 0010-4485

This paper applies metric-based mesh adaptation methods to advection-dominated tracer transport modelling problems in two and three dimensions, using the finite element package Firedrake. In particular, the mesh adaptation methods considered are built upon goal-oriented estimates for the error incurred in evaluating a diagnostic quantity of interest (QoI). In the motivating example of modelling to support desalination plant outfall design, such a QoI could be the salinity at the plant inlet, which could be negatively impacted by the transport of brine from the plant’s outfall. Four approaches are considered, one of which yields isotropic meshes. The focus on advection-dominated problems means that flows are often anisotropic; thus, three anisotropic approaches are also considered. Meshes resulting from each of the four approaches yield solutions to the tracer transport problem which give better approximations to QoI values than uniform meshing, for a given mesh size. The methodology is validated using an existing 2D tracer transport test case with a known analytical solution. Goal-oriented meshes for an idealised time-dependent desalination outfall scenario are also presented.

Journal article

Wallwork J, Barral N, Ham D, Piggott Met al., 2022, Goal-Oriented Error Estimation and Mesh Adaptation for Tracer Transport Modelling

<jats:p>This paper applies metric-based mesh adaptation methods to advection-dominated tracer transport modelling problems in two and three dimensions, using the finite element package Firedrake. In particular, the mesh adaptation methods considered are built upon goal-oriented estimates for the error incurred in evaluating a diagnostic quantity of interest (QoI). In the motivating example of modelling to support desalination plant outfall design, such a QoI could be the salinity at the plant inlet, which could be negatively impacted by the transport of brine from the plant's outfall. Four approaches are considered, one of which yields isotropic meshes. The focus on advection-dominated problems means that flows are often anisotropic; thus, three anisotropic approaches are also considered. Meshes resulting from each of the four approaches yield solutions to the tracer transport problem which give better approximations to QoI values than uniform meshing, for a given mesh size. The methodology is validated using an existing 2D tracer transport test case with a known analytical solution. Goal-oriented meshes for an idealised time-dependent desalination outfall scenario are also presented.</jats:p>

Working paper

Wallwork JG, Barral N, Kramer SC, Ham DA, Piggott MDet al., 2020, Goal-oriented error estimation and mesh adaptation for shallow water modelling, SN Applied Sciences, Vol: 2, Pages: 1-11, ISSN: 2523-3971

This study presents a novel goal-oriented error estimate for the nonlinear shallow water equations solved using a mixed discontinuous/continuous Galerkin approach. This error estimator takes account of the discontinuities in the discrete solution and is used to drive two metric-based mesh adaptation algorithms: one which yields isotropic meshes and another which yields anisotropic meshes. An implementation of these goal-oriented mesh adaptation algorithms is described, including a method for approximating the adjoint error term which arises in the error estimate. Results are presented for simulations of two model tidal farm configurations computed using the Thetis coastal ocean model (Kärnä et al. in Geosci Model Dev 11(11):4359–4382, 2018). Convergence analysis indicates that meshes resulting from the goal-oriented adaptation strategies permit accurate QoI estimation using fewer computational resources than uniform refinement.

Journal article

Wallwork JG, Barral N, Ham DA, Piggott MDet al., 2020, Anisotropic goal-oriented mesh adaptation in firedrake, IMR28 2019, Pages: 83-100

We consider metric-based mesh adaptation methods for steady-state partial differential equations (PDEs), solved using the finite element method in Firedrake. In this work, a number of mesh-adaptive methods are implemented within this framework, each enabling accurate approximation of a scalar quantity of interest (QoI). Through the QoI we define adjoint equations, with which we may gain understanding of its sensitivities to aspects of the PDE solution. Dual weighted residual type error estimation techniques are utilised in order to enable a goal-oriented strategy. Isotropic and anisotropic approaches are considered, both of which are able to achieve the same relative error in approximating the QoI as with uniform refinement, but using fewer elements. For validation purposes, we compare QoI values resulting from these approaches against analytical values which may be extracted for a particular advection-diffiusion based test case. Potential applications in desalination plant outfall modelling are discussed.

Conference paper

Wallwork J, Barral N, Kramer S, Ham D, Piggott Met al., 2019, Goal-Oriented Error Estimation and Mesh Adaptation for Shallow Water Modelling

Working paper

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