A Nash-Hormander iteration and boundary elements for the Molodensky problem

We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after m steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.Comment: 32 pages, 14 figures, to appear in Numerische Mathemati

Analysis of hollow inclusion–matrix debonding in particulate composites

AbstractThis work aims at understanding the effect of particle–matrix interfacial debonding on the tensile response of syntactic foams. The problem of a single hollow inclusion with spherical-cap cracks embedded in a dissimilar matrix material is studied. Degradation of elastic modulus, cavity formation in the proximity of debonded regions, stress localization phenomena in the inclusion, debonding energetics, and crack kinking are studied for a broad range of inclusion wall thickness and debonding extent. A series solution based on the Galerkin method is proposed and validated through comparison with findings from boundary element and finite element methods. Results are specialized to glass particle-vinyl ester matrix systems widely used in marine structural applications. The insight gained into the role of particle–matrix debonding extent and inclusion wall thickness is useful in understanding the possible failure mechanisms of syntactic foams under tensile and flexural loading conditions and in tailoring their parameters for specific applications

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

AbstractIn this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials

A Theoretical Analysis of Compactness of the Light Transport Operator

International audienceRendering photorealistic visuals of virtual scenes requires tractable models for the simulation of light. The rendering equation describes one such model using an integral equation, the crux of which is a continuous integral operator. A majority of rendering algorithms aim to approximate the effect of this light transport operator via discretization (using rays, particles, patches, etc.). Research spanning four decades has uncovered interesting properties and intuition surrounding this operator. In this paper we analyze compactness, a key property that is independent of its discretization and which characterizes the ability to approximate the operator uniformly by a sequence of finite rank operators. We conclusively prove lingering suspicions that this operator is not compact and therefore that any discretization that relies on a finite-rank or nonadaptive finite-bases is susceptible to unbounded error over arbitrary light distributions. Our result justifies the expectation for rendering algorithms to be evaluated using a variety of scenes and illumination conditions. We also discover that its lower dimensional counterpart (over purely diffuse scenes) is not compact except in special cases, and uncover connections with it being noninvertible and acting as a low-pass filter. We explain the relevance of our results in the context of previous work. We believe that our theoretical results will inform future rendering algorithms regarding practical choices.Le rendu d'images photoréalistes de scènes virtuelles nécessite la simulation du transport lumineux. L'équation du rendu décrit un tel modèle à l'aide d'une équation intégrale, ou intervient un opérateur intégral continu. Une part significative des d'algorithmes de rendu visent à approximer l'effet de cet opérateur via une discrétisation (à l'aide de rayons, de particules, de patchs, etc.). Quatre décennies de recherches ont mis à jour des propriétés et une intuition entourant cet opérateur. Dans cet article, nous analysons sa compacité, une propriété clé qui est indépendante de la discrétisation et qui caractérise la possibilité d'approcher uniformément l'opérateur par une suite d'opérateurs de rang fini. Nous justifions les soupçons persistants que cet opérateur n'est pas compact et donc que toute discrétisation qui repose sur un rang fini ou des bases finies non adaptatives n'apporte pas de guarantie d'erreur sur des distributions de lumière arbitraires. Notre résultat justifie le besoin d'évaluer chaque méthode en utilisant une variété de scènes et de conditions d'éclairage. Nous montrons également que son homologue de dimension inférieure (sur des scènes purement diffuses) n'est pas compact sauf dans des cas particuliers, et établissons un lien avec le fait qu'il est non inversible et agit comme un filtre passe-bas. Nous expliquons la pertinence de nos résultats dans le contexte de travaux antérieurs. Nous pensons que nos résultats théoriques éclaireront les futurs algorithmes de rendu concernant les choix pratiques

Numerical Study on the Magnetohydrodynamics of a Liquid Metal Oscillatory Flow under Inductionless Approximation

A harmonically-driven, incompressible, electrically conducting, and viscous liquid metal magnetohydrodynamic flow through a thin walled duct of rectangular cross section interacting with a uniform magnetic field traverse to its motion direction is numerically investigated. Chebyshev spectral collocation method is used to solve the Navier-Stokes equation under the inductionless approximation for the magnetic field in the gradient formulation for the electric field. Flow is considered fully developed in the direction perpendicular to the applied magnetic field and laminar in regime. Validation of numerical calculations respect to analytical calculations is established. Flow structure and key magnetohydrodynamic features regarding eventual alternating power generation application in a rectangular channel liquid metal magnetohydrodynamic generator setup are numerically inquired. Influence of pertinent parameters such as Hartmann number, oscillatory interaction parameter and wall conductance ratio on magnetohydrodynamic flow characteristics is illustrated. Particularly, it is found that in the side layer and its vicinity the emerging flow structures/patterns depend mainly on the Hartmann number and oscillatory interaction parameter ratio, while the situation for the Hartmann layer and its vicinity is less eventful. A similar feature has been discussed in the literature for the steady liquid metal flow case and served as rationale for developing the composite core-side-layer approximation to study the magnetohydrodynamics of liquid metal flows usable in direct power generation. In this study that approximation is not considered and the analysis is performed on liquid metal oscillatory (i., e., unsteady) flows usable in alternating power generation. Conversely, in terms of prospective practical applicability the formulation developed and tested with these calculations admits the implementation of a load resistance and walls conductivity optimization. That means that besides representing a numerical study on the magnetohydrodynamics of the oscillatory flow under consideration, absent in the literature for the parametric ranges reported, the formulation presently implemented can also be applicable to study the performance of an alternating liquid metal magnetohydrodynamic generator in the rectangular channel configuration