Adaptive meshless centres and RBF stencils for Poisson equation

We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method

Design Optimization for Spatial Arrangement of Used Nuclear Fuel Containers

Canada's proposed deep geological repository is a multiple-barrier system designed to isolate used nuclear fuel containers (UFCs) indefinitely with no release of radionuclides for at least one million years. Placing UFCs together as densely as possible is ideal for mitigating repository size and cost. However, due to heat generation from radioactive decay and material limitations, a key design criterion is that the maximum temperature inside the repository must not exceed 100 °C. To satisfy that criterion, design optimization for the spatial arrangement of UFCs in a crystalline rock repository is performed. Spatial arrangement pertains to: (i) the spacing between UFCs, (ii) the separation between placement rooms underground, and (iii) the locations of variously aged UFCs that generate heat at different rates. Most studies have considered UFCs to be identical in age during placement into the repository. Parameter analyses have also been performed to evaluate repository performance under probable geological conditions. In this work, the various ages of UFCs and the uncertainties in spacing-related design variables are of focus. Techniques for the actual placement of UFCs in the deep geological repository based on their age and methods for repository risk analysis using yield optimization are developed. The thermal evolution inside the deep geological repository is simulated using a finite element model. With many components inside the massive repository planned for upwards of 95,000 UFCs, direct optimization of the model is impractical or even infeasible due to it being computationally expensive to evaluate. Surrogate optimization is used to overcome that burden by reducing the number of detailed evaluations required to reach the optimal designs. Two placement cases are studied: (i) UFCs all having been discharged from a Canadian Deuterium Uranium reactor for 30 years, which is a worst-case scenario, and (ii) UFCs having been discharged between 30 and 60 years. Design options that have UFC spacing 1–2 m and placement room separation 10–40 m are explored. The placement locations of the variously aged UFCs are specified using either sinusoidal (cosine) functions or Kumaraswamy probability density functions. Yield optimization under assumed design variable tolerances and distributions is performed to minimize the probability of a system failure, which occurs when the maximum temperature constraint of 100 °C is exceeded. This method allows variabilities from the manufacturing and construction of the repository components that affect the design variables to be taken into account, incorporating a stochastic aspect into the design optimization that surrogate optimization would not include. Several distributions for the design variables are surveyed, and these include uniform, normal, and skewed distributions—all of which are approximated by Kumaraswamy distributions

Interpolating point spread function anisotropy

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require very accurate modeling and correction. While numerous methods have been devised to carry out the PSF correction itself, modeling of the PSF shape and its spatial variations across the instrument field of view has, so far, attracted much less attention. This step is nevertheless crucial because the PSF is only known at star positions while the correction has to be performed at any position on the sky. A reliable interpolation scheme is therefore mandatory and a popular approach has been to use low-order bivariate polynomials. In the present paper, we evaluate four other classical spatial interpolation methods based on splines (B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and ordinary Kriging (OK). These methods are tested on the Star-challenge part of the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and are compared with the classical polynomial fitting (Polyfit). We also test all our interpolation methods independently of the way the PSF is modeled, by interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are known exactly at star positions). We find in that case RBF to be the clear winner, closely followed by the other local methods, IDW and OK. The global methods, Polyfit and B-splines, are largely behind, especially in fields with (ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all interpolators reach a variance on PSF systematics $\sigma_{sys}^2$ better than the $1\times10^{-7}$ upper bound expected by future space-based surveys, with the local interpolators performing better than the global ones

Learning feedforward controller for a mobile robot vehicle

This paper describes the design and realisation of an on-line learning posetracking controller for a three-wheeled mobile robot vehicle. The controller consists of two components. The first is a constant-gain feedback component, designed on the basis of a second-order model. The second is a learning feedforward component, containing a single-layer neural network, that generates a control contribution on the basis of the desired trajectory of the vehicle. The neural network uses B-spline basis functions, enabling a computationally fast implementation and fast learning. The resulting control system is able to correct for errors due to parameter mismatches and classes of structural errors in the model used for the controller design. After sufficient learning, an existing static gain controller designed on the basis of an extensive model has been outperformed in terms of tracking accuracy

Development of numerical and data models for the support of digital twins in offshore wind engineering

Error on title page. Date of award is 2022.As offshore wind farms grow there is a continued demand for reduced costs. Maintenance costs and downtime can be reduced through greater information on the asset in relation to its operational loads and structural resistance to damage and so there is an increasing interest in digital twin technologies. Through digital twins, an operational asset can be replicated computationally, thus providing more information. Modelling these aspects requires a wide variety of models in different fields. To advance the feasibility of digital twin technology this thesis aims to develop the multi-disciplinary set of modelling domains which help form the basis of future digital twins. Throughout this work, results have been validated against operational data recorded from sensors on offshore structures. This has provided value and confidence to the results as it shows how well the mix of state-of-the art models compare to real world engineering systems. This research presents a portfolio of five research areas which have been published in a mix of peer-reviewed journal articles and conference papers. These areas are: 1) A computational fluid dynamics (CFD) model of an offshore wind farm conducted using a modified solver in the opensource software. This work implements actuator disk turbine models and uses Reynolds averaged Naiver Stokes approaches to represent the turbulence. This investigates the impact of modelling choices and demonstrates the impact of varied model parameters. The results are compared to operational site data and the modelling errors are quantified. There is good agreement between the models and site data. 2) An expansion on traditional CFD approaches through incorporating machine learning (ML). These ML models are used to approximate the results of the CFD and thereby allow for further analysis which retains the fidelity of CFD at comparatively negligible computational cost. The results are compared to operational site data and the errors at each step are quantified for validation. 3) A time-series forecasting of weather variables based on past measured data. A novel approach for forecasting time-series is developed and compared to two existing methods: Markov-Chains and Gradient Boosting. While this new method is more complex and requires more time to train, it has the desirable feature of incorporating seasonality at multiple timescales and thus providing a more representative time-series. 4) An investigation of the change in modal parameters in an offshore wind jacket structure from damages or from changing operational conditions. In this work the detailed design model of the structure from Ramboll is used. This section relates the measurable modal parameters to the operational condition through a modelling approach. 5) A study conducted using accelerometer data from an Offshore Substation located in a wind farm site. Operational data from 12 accelerometers is used to investigate the efficacy of several potential sensor layouts and therefore to quantify the consequence of placement decisions. The results of these developments are an overall improvement in the modelling approaches necessary towards the realisation of digital twins as well as useful development in each of the component areas. Both areas related to wind loading as well as structural dynamics have been related to operational data. The validation of this link between the measured and the modelled domains facilitates operators and those in maintenance in gaining more information and greater insights into the conditions of their assets.As offshore wind farms grow there is a continued demand for reduced costs. Maintenance costs and downtime can be reduced through greater information on the asset in relation to its operational loads and structural resistance to damage and so there is an increasing interest in digital twin technologies. Through digital twins, an operational asset can be replicated computationally, thus providing more information. Modelling these aspects requires a wide variety of models in different fields. To advance the feasibility of digital twin technology this thesis aims to develop the multi-disciplinary set of modelling domains which help form the basis of future digital twins. Throughout this work, results have been validated against operational data recorded from sensors on offshore structures. This has provided value and confidence to the results as it shows how well the mix of state-of-the art models compare to real world engineering systems. This research presents a portfolio of five research areas which have been published in a mix of peer-reviewed journal articles and conference papers. These areas are: 1) A computational fluid dynamics (CFD) model of an offshore wind farm conducted using a modified solver in the opensource software. This work implements actuator disk turbine models and uses Reynolds averaged Naiver Stokes approaches to represent the turbulence. This investigates the impact of modelling choices and demonstrates the impact of varied model parameters. The results are compared to operational site data and the modelling errors are quantified. There is good agreement between the models and site data. 2) An expansion on traditional CFD approaches through incorporating machine learning (ML). These ML models are used to approximate the results of the CFD and thereby allow for further analysis which retains the fidelity of CFD at comparatively negligible computational cost. The results are compared to operational site data and the errors at each step are quantified for validation. 3) A time-series forecasting of weather variables based on past measured data. A novel approach for forecasting time-series is developed and compared to two existing methods: Markov-Chains and Gradient Boosting. While this new method is more complex and requires more time to train, it has the desirable feature of incorporating seasonality at multiple timescales and thus providing a more representative time-series. 4) An investigation of the change in modal parameters in an offshore wind jacket structure from damages or from changing operational conditions. In this work the detailed design model of the structure from Ramboll is used. This section relates the measurable modal parameters to the operational condition through a modelling approach. 5) A study conducted using accelerometer data from an Offshore Substation located in a wind farm site. Operational data from 12 accelerometers is used to investigate the efficacy of several potential sensor layouts and therefore to quantify the consequence of placement decisions. The results of these developments are an overall improvement in the modelling approaches necessary towards the realisation of digital twins as well as useful development in each of the component areas. Both areas related to wind loading as well as structural dynamics have been related to operational data. The validation of this link between the measured and the modelled domains facilitates operators and those in maintenance in gaining more information and greater insights into the conditions of their assets

Doctor of Philosophy

dissertationPlatelet aggregation, an important part of the development of blood clots, is a complex process involving both mechanical interaction between platelets and blood, and chemical transport on and o the surfaces of those platelets. Radial Basis Function (RBF) interpolation is a meshfree method for the interpolation of multidimensional scattered data, and therefore well-suited for the development of meshfree numerical methods. This dissertation explores the use of RBF interpolation for the simulation of both the chemistry and mechanics of platelet aggregation. We rst develop a parametric RBF representation for closed platelet surfaces represented by scattered nodes in both two and three dimensions. We compare this new RBF model to Fourier models in terms of computational cost and errors in shape representation. We then augment the Immersed Boundary (IB) method, a method for uid-structure interaction, with our RBF geometric model. We apply the resultant method to a simulation of platelet aggregation, and present comparisons against the traditional IB method. We next consider a two-dimensional problem where platelets are suspended in a stationary fluid, with chemical diusion in the fluid and chemical reaction-diusion on platelet surfaces. To tackle the latter, we propose a new method based on RBF-generated nite dierences (RBF-FD) for solving partial dierential equations (PDEs) on surfaces embedded in 2D domains. To robustly tackle the former, we remove a limitation of the Augmented Forcing method (AFM), a method for solving PDEs on domains containing curved objects, using RBF-based symmetric Hermite interpolation. Next, we extend our RBF-FD method to the numerical solution of PDEs on surfaces embedded in 3D domains, proposing a new method of stabilizing RBF-FD discretizations on surfaces. We perform convergence studies and present applications motivated by biology. We conclude with a summary of the thesis research and present an overview of future research directions, including spectrally-accurate projection methods, an extension of the Regularized Stokeslet method, RBF-FD for variable-coecient diusion, and boundary conditions for RBF-FD

Efficient Numerical Methods for Pricing American Options under Lévy Models

Two new numerical methods for the valuation of American and Bermudan options are proposed, which admit a large class of asset price models for the underlying. In particular, the methods can be applied with Lévy models that admit jumps in the asset price. These models provide a more realistic description of market prices and lead to better calibration results than the well-known Black-Scholes model. The proposed methods are not based on the indirect approach via partial differential equations, but directly compute option prices as risk-neutral expectation values. The expectation values are approximated by numerical quadrature methods. While this approach is initially limited to European options, the proposed combination with interpolation methods also allows for pricing of Bermudan and American options. Two different interpolation methods are used. These are cubic splines on the one hand and a mesh-free interpolation by radial basis functions on the other hand. The resulting valuation methods allow for an adaptive space discretization and error control. Their numerical properties are analyzed and, finally, the methods are validated and tested against various single-asset and multi-asset options under different market models

Accurate Stabilization Techniques for RBF-FD Meshless Discretizations with Neumann Boundary Conditions

A major obstacle to the application of the standard Radial Basis Function-generated Finite Difference (RBF-FD) meshless method is constituted by its inability to accurately and consistently solve boundary value problems involving Neumann boundary conditions (BCs). This is also due to ill-conditioning issues affecting the interpolation matrix when boundary derivatives are imposed in strong form. In this paper these ill-conditioning issues and subsequent instabilities affecting the application of the RBF-FD method in presence of Neumann BCs are analyzed both theoretically and numerically. The theoretical motivations for the onset of such issues are derived by highlighting the dependence of the determinant of the local interpolation matrix upon the boundary normals. Qualitative investigations are also carried out numerically by studying a reference stencil and looking for correlations between its geometry and the properties of the associated interpolation matrix. Based on the previous analyses, two approaches are derived to overcome the initial problem. The corresponding stabilization properties are finally assessed by succesfully applying such approaches to the stabilization of the Helmholtz-Hodge decomposition