Numerical Analysis and Spanwise Shape Optimization for Finite Wings of Arbitrary Aspect Ratio

This work focuses on the development of efficient methods for wing shape optimization for morphing wing technologies. Existing wing shape optimization processes typically rely on computational fluid dynamics tools for aerodynamic analysis, but the computational cost of these tools makes optimization of all but the most basic problems intractable. In this work, we present a set of tools that can be used to efficiently explore the design spaces of morphing wings without reducing the fidelity of the results significantly. Specifically, this work discusses automatic differentiation of an aerodynamic analysis tool based on lifting line theory, a light-weight gradient-based optimization framework that provides a parallel function evaluation capability not found in similar frameworks, and a modification to the lifting line equations that makes the analysis method and optimization process suitable to wings of arbitrary aspect ratio. The toolset discussed is applied to several wing shape optimization problems. Additionally, a method for visualizing the design space of a morphing wing using this toolset is presented. As a result of this work, a light-weight wing shape optimization method is available for analysis of morphing wing designs that reduces the computational cost by several orders of magnitude over traditional methods without significantly reducing the accuracy of the results

Power Integral Points on Elliptic Curves

This thesis looks at some of the modern approaches towards the solution of Diophantine equations, and utilizes them to display the nonexistence of perfect powers occurring in certain types of sequences. In particular we look at the denominator divisibility sequences (Bn) formed by Mordell elliptic curves ED : y2 = x3+D. For the curve-point pair (E−2, P), where E−2 : y2 = x3 −2, and P = (3, 5) is a nontorsion point, we prove that no term Bn is a perfect 5th power, and we give the explicit bound p � 137 for any term in the associated elliptic denominator sequence to be a perfect power Bn = Zpn for 1 < n < 113762879. We then look at obtaining upper bounds on p for the seventy-two rank 1 Mordell curves in the range |D| < 200 to possess a pth perfect power. This is done by consideration of the finite number of rational and irrational newforms corresponding to an also finite number of levels of these newforms: in thirty cases we give a bound via examination of both the rational and irrational cases, and for the remaining forty-two cases our bound is merely for the rational case due to computational limitations

Nonintrusive parametric solutions in structural dynamics

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A nonintrusive reduced order method able to solve a parametric modal analysis is proposed in this work. The main objective is being able to efficiently identify how a variation of user-defined parameters affects the dynamic response of the structure in terms of fundamental natural frequencies and corresponding mode shapes. A parametric version of the inverse power method (IPM) is presented by using the proper generalised decomposition (PGD) rationale. The proposed approach utilises the socalled encapsulated PGD toolbox and includes a new algorithm for computing the square root of a parametric object. With only one offline computation, the proposed PGD-IPM approach provides an analytical parametric expression of the smallest (in magnitude) eigenvalue (or natural frequency) and corresponding eigenvector (mode shape), which contains all the possible solutions for every combination of the parameters within pre-defined ranges. A Lagrange multiplier deflation technique is introduced in order to compute subsequent eigenpairs, which is also valid to overcome the stiffness matrix singularity in the case of a free-free structure. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. Two numerical examples are shown to underline the properties of the technique. The first example includes one material and one geometric parameter. The second example shows a more realistic industrial example, where the nonintrusivity of the approach is demonstrated by employing a commercial FE package for assembling the FE matrices. Finally, a multi-objective optimisation study is performed proving that the developed method could significantly assist the decision-making during the preliminary phase of a new design process.This project is part of the Marie Skłodowska-Curie ITN-EJD ProTechTion funded by the European Union Horizon 2020 research and innovation program with Grant Number 764636. The work of Fabiola Cavaliere, Sergio Zlotnik and Pedro Díez is partially supported by the MCIN/AEI/10.13039/501100011033, Spain (Grant Number: PID2020-113463RB-C32, PID2020-113463RB-C33 and CEX2018-000797-S). Ruben Sevilla also acknowledges the support of the Engineering and Physical Sciences Research Council (Grant Number: EP/P033997/1).Peer ReviewedPostprint (author's final draft

Asymptotic theory for Bayesian nonparametric inference in statistical models arising from partial differential equations

Partial differential equations (PDEs) are primary mathematical tools to model the behaviour of complex real-world systems. PDEs generally include a collection of parameters in their formulation, which are often unknown in applications and need to be estimated from the data. In the present thesis, we investigate the theoretical performance of nonparametric Bayesian procedures in such parameter identification problems in PDEs. In particular, inverse regression models for elliptic equations and stochastic diffusion models are considered. In Chapter 2, we study the statistical inverse problem of recovering an unknown function from a linear indirect measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein–von Mises theorem for a large collection of linear functionals of the unknown, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The general result is applied to three concrete examples that cover both the mildly and severely ill-posed cases: specifically, elliptic inverse problems, an elliptic boundary value problem, and the recovery of the initial condition of the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser is an efficient estimator, and we derive frequentist guarantees for certain credible balls centred around it. Chapter 3 is concerned with statistical nonlinear inverse problems. We focus on the prototypical example of recovering the unknown conductivity function in an elliptic PDE in divergence form from discrete noisy point evaluations of the PDE solution. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate, algebraic in inverse sample size, for the estimation error of the associated posterior means. Finally, in Chapter 4 we extend the posterior consistency analysis to dynamical models based on stochastic differential equations. We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension. Chapter 1 is dedicated to introducing the statistical models considered in Chapters 2 - 4, and to providing an overview of the theoretical results derived therein. The main theorems of Chapter 2 and Chapter 3 are illustrated via the results of simulations, and detailed comments are provided on the implementation.Richard Nickl’s ERC grant No. 647812; EPSRC grant EP/L016516/1 for the Cambridge Centre for Analysi

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.