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In this paper we provide a complete theoretical analysis of a two-dimensional degenerate nonconvex singular stochastic control problem. The optimisation is motivated by a storage-consumption model in an electricity market, and features a stochastic real-valued spot price modelled by Brownian motion. We find analytical expressions for the value function, the optimal control, and the boundaries of the action and inaction regions. The optimal policy is characterised in terms of two monotone and discontinuous repelling free boundaries, although part of one boundary is constant and the smooth fit condition holds there. Funding: The first and the third authors were supported by the Engineering and Physical Sciences Research Council (EPSRC) [Grant EP/K00557X/1]. Financial support by the German Research Foundation (DFG) through the Collaborative Research Centre 'Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications' is gratefully acknowledged by the second author. Keywords: finite-fuel singular stochastic control * optimal stopping * free boundary * Hamilton-Jacobi-Bellman equation * irreversible investment * electricity market, 1. Introduction Consider the following problem introduced in De Angelis et al. [4]: a firm purchases electricity over time at a stochastic spot price [([X.sub.t]).sub.t[greater than or equal to]0] for [...]

We revisit lower bounds on the regret in the case of multiarmed bandit problems. We obtain nonasymptotic, distribution-dependent bounds and provide simple proofs based only on well-known properties of Kullback-Leibler divergences. These bounds show in particular that in the initial phase the regret grows almost linearly, and that the well-known logarithmic growth of the regret only holds in a final phase. The proof techniques come to the essence of the information-theoretic arguments used and they involve no unnecessary complications. Funding: This work was partially supported by the CIMI (Centre International de Mathdmatiques et d'Informatique) Excellence program while Gilles Stoltz visited Toulouse in November 2015. The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under project SPADRO [Grant ANR-13-BS01-0005] and under project ALICIA [Grant ANR-13-CORD-0020]. Gilles Stoltz would like to thank Investissements d'Avenir [Grant ANR-11-IDEX-0003/Labex Ecodec/ANR-ll-LABX-0047] for financial support. Keywords: multiarmed bandits * cumulative regret * Information-theoretic proof techniques * nonasymptotic lower bounds, 1. Introduction After the works of Lai and Robbins [21] and Burnetas and Katehakis [9], it is widely admitted that the growth of the cumulative regret in a bandit problem [...]

We consider a statistical inverse learning (also called inverse regression) problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points [Formula omitted], superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependency of the constant factor in the variance of the noise and the radius of the source condition set., Author(s): Gilles Blanchard [sup.1] , Nicole Mücke [sup.1] Author Affiliations: (Aff1) 0000 0001 0942 1117, grid.11348.3f, Institute of Mathematics, University of Potsdam, , Karl-Liebknecht-Strasse 24-25, 14476, Potsdam, Germany Introduction Setting [...]

Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e., while preserving the length of curves, in a twice differentiable way. An unexpected result by Nash (Ann Math 60:383-396, 1954 (See CR18)) and Kuiper (Indag Math 17:545-555, 1955 (See CR14)) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere, the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here, we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a nonlinear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a [Formula omitted] fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the [Formula omitted] fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment., Author(s): Evangelis Bartzos [sup.1] , Vincent Borrelli [sup.2] , Roland Denis [sup.3] , Francis Lazarus [sup.4] , Damien Rohmer [sup.5] , Boris Thibert [sup.6] Author Affiliations: (Aff1) 0000 0001 2155 [...]

We prove exponential rates of convergence of hp-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet-Neumann boundary conditions and to anisotropic, which increase linearly over mesh layers away from edges and vertices. In particular, we construct [Formula omitted]-conforming quasi-interpolation operators with N degrees of freedom and prove exponential consistency bounds [Formula omitted] for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on countably normed classes of weighted Sobolev spaces with non-homogeneous weights in the vicinity of Neumann edges., Author(s): Dominik Schötzau [sup.1] , Christoph Schwab [sup.2] Author Affiliations: (Aff1) 0000 0001 2288 9830, grid.17091.3e, Mathematics Department, University of British Columbia, , 1984 Mathematics Road, V6T 1Z2, Vancouver, BC, [...]

Exact non-reflecting boundary conditions for a linear incompletely parabolic system in one dimension have been studied. The system is a model for the linearized compressible Navier-Stokes equations, but is less complicated which allows for a detailed analysis without approximations. It is shown that well-posedness is a fundamental property of the exact non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, it is also shown that energy stability follows automatically., Author(s): Sofia Eriksson [sup.1] , Jan Nordström [sup.2] Author Affiliations: (1) Department of Mathematics, Technische Universität Darmstadt, 0000 0001 0940 1669, grid.6546.1, , Dolivostr. 15, 64293, Darmstadt, Germany (2) Department [...]

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples., Author(s): Martin Burger [sup.1] [sup.2] , Ole Løseth Elvetun [sup.3] , Matthias Schlottbom [sup.1] Author Affiliations: (1) 0000 0001 2172 9288grid.5949.1Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. [...]

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's first theorem (1918 (See CR6)). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional., Author(s): Elizabeth L. Mansfield [sup.1] , Tristan Pryer [sup.2] Author Affiliations: (1) 0000 0001 2232 2818grid.9759.2School of Mathematics, Statistics and Actuarial Science, University of Kent, CT2 7NF, Canterbury, UK (2) [...]

In this study, stochastic numerical treatment is presented for boundary value problems (BVPs) arising in nanofluidics for nonlinear Jeffery-Hamel flow (NJ-HF) equations using feed-forward artificial neural networks (ANNs) optimized with bio-inspired computing based on genetic algorithms (GAs) integrated with the active-set method (ASM). NJ-HF equations associated with both convergent and divergent channels, involving nanoparticles, are derived from the transformation of Navier- Stokes partial differential equations to nonlinear BVPs of third-order ordinary differential equations. The mathematical model of the transformed BVPs is developed with the help of ANNs in an unsupervised manner and the design parameters of these networks are trained with GAs, ASM, and GA-ASM. The design scheme is evaluated for NJ-HF by taking water as a base fluid containing three different types of nanomaterials: copper (Cu), alumina ([Al.sub.2][O.sub.3]), and titania (Ti[O.sub.2]) under various scenarios based on the angle of the channels and Reynolds numbers. Accuracy and convergence of the designed scheme are validated through comparison with standard numerical results using the Adams method. Key words: nanofludics, nonlinear Jeffery-Hamel flow, artificial neural networks, nanotechnology, genetic algorithm, active-set method. Nous presentons ici une etude numerique stochastique du probleme aux limites (BVP) qu'on retrouve avec les equations de l'ecoulement de Jeffery-Hamel (NJ-HF) pour un nanofluide non lineaire, en utilisant une boucle d'anticipation d'un reseau de neurones artificiels (ANN) optimise par un calcul bio-inspire base sur un algorithme genetique (GA) et integre par methode du set actif (ASM). Les equations de NJ-HF, associees aux canaux convergents et divergents impliquant des nanoparticules, sont obtenues de la transformation des equations de Navier-Stokes aux derivees partielles en BVP non lineaire d'equations differentielles ordinaires. Le modele mathematique de la transformation en BVP est developpe avec l'aide d'un ANN sans supervision et les parametres de conception de ce reseau sont developpes avec l'aide de GA, ASM et GA-ASM. Le schema de cette conception est evalue pour NJ-HF en prenant une base d'eau contenant trois types differents de nanomateriaux, du cuivre (Cu), de l'alumine ([Al.sub.2][O.sub.3]) et du bioxyde de titane (TiO2), sous differents scenarios selon l'angle des canaux et le nombre de Reynolds. La precision et la convergence sont validees par comparaison avec les resultats numeriques standard obtenus de la methode d'Adams. [Traduit par la Redaction] Mots-cles: nanofluide, ecoulement non lineaire de Jeffery-Hamel, reseau de neurones artificiels, nanotechnologie, algorithme genetique, methode du set actif. PACS Nos.: 02.30.Hq, 07.05.Mh, 85.85.+j., 1. Introduction Dynamics and heat transfer properties of fluids containing nanomaterials have been extensively investigated because of their diverse applications in applied science and technology [1-4]. Nanofluids generally consist of [...]

We study the worst case complexity and the tractability of the model second-order problem [Formula omitted] on [Formula omitted] with homogeneous Neumann boundary conditions. As is often the case, we study the variational formulation of this problem. Previous work on the tractability of problems such as this relied on the fact that such problems were reducible to the [Formula omitted]-approximation problem, which allowed us to find necessary and sufficient conditions for the problem to have a given degree of tractability. However, such an approach can only yield sufficient conditions, and not necessary conditions, for the model second-order Neumann problem to have a given degree of tractability. In this paper, we remedy this gap and find necessary and sufficient conditions for this Neumann problem to exhibit a given degree of tractability., Author(s): A. G. Werschulz[sup.1] [sup.2] , H. Wozniakowski[sup.2] [sup.3] Author Affiliations: (1) Department of Computer and Information Sciences, Fordham University, 10023, New York, NY , USA (2) Department of Computer [...]

Distribution of the electric potential in a very long plate (for example, a long metal ruler) is determined. This is achieved by conformally mapping the plate into a plane, simplifying the geometry of the boundary conditions. Singularities of the potential are discussed as well as their regularization by the final size of electrical contacts. An analogy with renormalization is pointed out. Results are compared with previous studies. PACS Nos.: 01.55.+b, 03.50.De. Nous determinons la distribution du potentiel electrique dans une plaque tres longue (comme une regle metallique). Par transformation conforme, la regle devient un plan, simplifiant la geometrie des conditions limites. Nous analysons les singularites du potentiel, ainsi que leur regularisation suivant la dimension finie des contacts. Nous soulignons une analogie avec la renormalisation. Nous comparons nos resultats avec des etudes anterieures. [Traduit par la Redaction], 1. Introduction This paper is motivated by an intriguing question posed in an excellent collection of physics problems [1], based on an exam problem from the prestigious Moscow Institute of [...]

We consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the [...]

W. Magnus introduced a particular differential equation characterizing the logarithm of the solution of linear initial value problems for linear operators. The recursive solution of this differential equation leads to a peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli numbers, iterated Lie brackets and integrals. This paper aims at obtaining further insights into the fine structure of the Magnus expansion. By using basic combinatorics on planar rooted trees we prove a closed formula for the Magnus expansion in the context of free dendriform algebra. From this, by using a well-known dendriform algebra structure on the vector space generated by the disjoint union of the symmetric groups, we derive the Mielnik-Plebariski-Strichartz formula for the continuous Baker-Campbell-Hausdorff series. Keywords Magnus expansion * B-Series * Trees * Pre-Lie algebra * Dendriform algebra * Rota-Baxter algebra * Permutations Mathematics Subject Classification 45A05 * 05C05 * 17D25, 1 Introduction During the last decade, a surprising convergence of different areas of mathematical sciences has occurred. The seemingly separate fields: numerical integration methods, Lyons' rough path theory [23, 37], [...]

We investigate the twin paradox assuming the acceleration acts instantaneously on one of the twins and that its effect is just to revert the relative movement of the twins keeping the same relative speed. The relative motion of the twins is then split in two stages: one where they move away and another when they approach each other. Each stage is described by specific Lorentz transformations that obey certain boundary conditions related to the reversion of motion. We then show how the paradox arises from the particular form of the Lorentz transformation describing the approaching movement of the twins. PACS No: 03.30.+p Nous analysons le paradoxe des jumeaux en supposant que l'acceleration agit instantanement sur l'un des jumeaux et ne fait que renverser le mouvement relatif des jumeaux sans changer la grandeur de la vitesse. Le mouvement relatif des jumeaux est alors divise en deux etapes : l'une de separation et l'autre de rapprochement. Chaque etape est decrite par des transformations specifiques de Lorentz qui obeissent a certaines conditions limites reliees au renversement du mouvement. Nous montrons alors comment le paradoxe nait de la forme particuliere de la transformation de Lorentz qui decrit le mouvement lorsque les jumeaux se rapprochent. [Traduit par la Redaction], 1. Introduction The validity of the theory of special relativity (SR) is strongly enforced by many experiments in high-energy physics which, so far, have shown no signs of violation of [...]

Griffiths proposed a pair of boundary conditions that define a point interaction in one dimensional quantum mechanics. The conditions involve the nth derivative of the wave function where n is a non-negative integer. We re-examine the interaction so defined and explicitly confirm that it is self-adjoint for any even value of n and for n = 1. The interaction is not self-adjoint for odd n > 1. We then propose a similar but different pair of boundary conditions with the nth derivative of the wave function such that the ensuing point interaction is self-adjoint for any value of n. PACS Nos: 03.65.-w Griffiths a propose une paire de conditions a une limite qui definissent une interaction ponctuelle en mecanique quantique en une dimension. La condition implique la ne derivee de la fonction d'onde, ou n est un entier positif. Nous reexaminons l'interaction ainsi definie et confirmons explicitement qu'elle est self adjointe pour toute valeur paire de n et pour n =1. Elle n'est pas self adjointe pour les valeurs de n > 1 impaires. Nous proposons alors une paire de conditions similaires, mais differentes, avec la ne derivee de la fonction d'onde telle que le point d'interaction qui s'ensuit est self adjoint pour toute valeur de n. [Traduit par la Redaction], 1. Introduction In one-dimensional quantum mechanics, the simplest point interaction is the one that is represented by the potential V(x) = [[[??].sup.2]/2m]c[delta](x) (1) where m is the mass of the [...]

The finite element method (FEM) has become a standard tool for solving complex problems in geotechnical engineering. Many different advanced constitutive models for fine-grained soils have been developed in recent years, which can consider various phenomena of soil. Generally, the number of constitutive parameters increases with phenomena incorporated in the model, and their determination is one of the key issues in numerical modeling in geotechnics. Normally, experimental data, experience, and back analyses based on engineering judgment are used to arrive at appropriate input parameters for a particular model. However, this procedure is not always satisfactory, especially when the number of required input parameters is large. In this paper, a population-based algorithm has been used to determine the constitutive parameters for a geotechnical boundary value problem, namely a floating stone column foundation under an embankment. Measurements of surface settlements and excess pore-water pressures at different depths are available for calibration. A subsequent statistical assessment of the calibration results is followed to assess the quality of the identified parameters in dependency of the different set of measurements evaluated. As a major result of this research, measures of the utility and reliability of the constitutive models for further predictive computations can be estimated. Key words: numerical modeling, clays, constitutive relations, creep, field instrumentation, ground improvement. La methode par elements finis (MEF) est devenue un outil standard pour solutionner des problemes complexes en genie geotechnique. Plusieurs modeles constitutifs avances differents, pour des sols a granulometrie fine et pouvant considerer des phenomenes varies dans les sols, ont ete developpes dans les dernieres annees. Generalement, le nombre de parametres constitutifs augmente selon les phenomenes incorpores dans le modele, et leur determination est un probleme important relie a la modelisation numerique en geotechnique. Normalement, les donnees experimentales, l'experience et les retro- analyses basees sur un jugement d'ingenierie sont utilisees pour arriver aux parametres d'entree appropries pour un modele donne. Cependant, cette procedure n'est pas toujours satisfaisante, particulierement lorsque le nombre de parametres d' entree requis est grand. Dans cet article, un algorithme base sur la population a ete utilise pour determiner les parametres constitutifs d'un probleme de valeur frontiere en geotechnique, soit une fondation faite d'une colonne en pierre flottante sous un remblai. Des mesures des tassements de surface et de la pression interstitielle excessive a differentes profondeurs sont disponibles pour le calibrage. Une analyse statistique subsequente des resultats du calibrage est realisee pour evaluer la qualite des parametres identifies en dependance aux differents ensembles de donnees evalues. Un resultat majeur de cette recherche est la possibilite d'estimer les mesures de l'utilite et de la fiabilite des modeles constitutifs pour des calculs predictifs supplementaires. Mots-cles : modelisation numerique, argiles, relations constitutives, fluage, instrumentation de terrain, amelioration du sol. [Traduit par la Redaction], Introduction There is an increasing demand for the application of advanced numerical methods in geotechnical engineering. Existing commercial finite element (FE) codes provide many different types of constitutive models to [...]

A complete second-order analytical solution for the nonlinear diffraction of waves by 2-D rectangular cylinders fixed at the free surface of finite-depth waters is presented. As a sequel to a previous linear treatment of the same problem, the present study revisits the nonlinear wave diffraction modelling in 2-D, in which a series of boundary value problems are built up by a classical perturbation expansion. Particular solution for the second-order potential is derived by means of a Fourier integral theorem, whilst the homogeneous second-order solution is obtained in a way similar to that of the linear problem. The present solution is validated by comparisons with experiments as well as with previous computational results of various researchers. Key words: 2-D wave diffraction, rectangular barriers, second-order harmonics. Une solution analytique complete de second ordre pour la diffraction non lineaire de vagues par des cylindres rectangulaires bidimensionnels fixes a la surface libre des eaux a profondeur finie est presentee. En tant que suite au traitement lineaire anterieur au meme probleme, l'etude present reexamine la modelisation non lineaire des vagues en deux dimensions, dans laquelle une serie de problemes aux limites est construite par une expansion classique des perturbations. Une solution particuliere pour le potentiel de second ordre est derivee en utilisant la transformee de Fourier, alors que la solution homogene de second ordre est obtenue par un moyen similaire a celui du probleme lineaire. La solution actuelle est validee par des comparaisons a des experiences ainsi que des resultats computationnels anterieurs provenant de divers chercheurs. Mots-cles : diffraction bidimensionnelle des vagues, barrieres rectangulaires, harmoniques de second ordre. [Traduit par la Redaction], 1. Introduction Versatility of linear solutions to the problem of wave radiation and diffraction by horizontal barriers at the free surface, makes them advantageous in most cases for engineering purposes. [...]

A coupled numerical model of flow through porous media, formulated in terms of the fluid displacements and changing the impermeability boundary conditions, is applied to calculate the free surface in unconfined, steady-seepage problems. In this new methodology, the domain of computation is kept constant, and the free surface is iteratively obtained by imposing impermeability conditions at the free boundaries, which are updated at the end of each iteration, aiming to simulate the physics of the problem. By doing so, a line of pore-water pressure equal to zero (free surface) is obtained in a few iterations, usually two or three. The results obtained with the proposed method have been successfully compared to analytical and numerical solutions in several examples to show the usefulness of the method in practical seepage problems. Key words: free surface, coupled models, steady flow, porous media. Un modele numerique couple d'ecoulement a travers un medium poreux, formule selon les deplacements des fluides et en changeant les conditions frontieres d'impermeabilite, est applique pour calculer la surface libre dans le cas de problemes de percolation stable et non confinee. Dans cette nouvelle technologie, le domaine de calcul demeure constant, et la surface libre est obtenue par iteration en imposant des conditions d'impermeabilite aux frontieres libres, qui sont mises a jour a la fin de chaque iteration simulant la physique du probleme. De cette facon, une ligne de pression interstitielle egale a zero (surface libre) est obtenue en quelques iterations, normalement deux ou trois. Les resultats obtenus avec la methode proposee ont ete compares avec succes a des solutions analytiques et numeriques provenant de plusieurs exemples afin de demontrer son utilite pour des problemes pratiques de percolation. Mots-cles: surface libre, modeles couples, ecoulement stable, medium poreux. [Traduit par la Redaction], Introduction For determining the free surface in unconfined flow through porous media, it is general practice to assume totally dry or saturated soil conditions. Following this line of thought, the [...]

A general first-order formulation of the continuous sensitivity equations is introduced that improves the accuracy of the sensitivity boundary conditions. In the continuous sensitivity method, design or shape parameter gradients are computed from a continuous system of partial differential equations instead of the discretized system, which avoids the problematic mesh sensitivity calculations of discrete methods for shape variation problems. The first-order formulation for both the underlying elasticity and sensitivity equations is amenable to solution by a high-order polynomial least-squares finite element model. The continuous sensitivity boundary-value problem, which can be posed in local or total derivative form, is generally simpler in local sensitivity form for shape variation problems. Although the local form is not unique and total material sensitivities are generally desired for structural elasticity problems, the local sensitivity solution is easily transformed to a material derivative. The first-order formulation and the transformation to material derivatives are demonstrated for two elasticity example problems. The least-squares continuous sensitivity solutions are presented and compared with analytic sensitivities and finite difference results and should prove convenient validation benchmarks for other continuous sensitivity applications. DOI: 10.2514/1.44349

Finite aquifer solution exists for the constant head in a fully penetrating well. Their use for well recharge is limited, as they do not permit simultaneous computation of unsteady wellhead pressure and variable recharge rate. In the present paper semi-analytical solutions are presented for well recharge under variable head boundary condition. These solutions were developed using the method of separation of variables and Duhamel's convolution theorem. The solution developed in the paper was verified with the Jacob-Lohman solution and subsequently validated using field data pertinent to constant-head boundary conditions. Subsequently for variable head boundary condition such an appropriate background was found missing in the literature. DOI: 10.1061/(ASCE)IR.1943-4774.0000255 CE Database subject headings: Aquifers: Wells; Water storage. Author keywords: Separation of variable method: Bessel-Fourier series: Finite aquifer; Discrete kernel; Well storage.

We study the problem of a nonrelativistic quantum particle moving on a real line with an idealized and localized singular interaction with zero range at x = 0 (i.e., a point interaction there). This kind of system can be described in two ways: (i) by considering an alternative free system (i.e., without the singular potential) but excluding the point x = 0 (In this case, the point interaction is exclusively encoded in the boundary conditions.) and (ii) by explicitly considering the singular interaction by means of a local singular potential. In this paper we relate, compare, and discuss, in a simple and pedagogical way these two equivalent approaches. Our main goal in this paper is to introduce the essential ideas about point interactions in a very accesible form to advanced undergraduates. PACS Nos: 03.65.-w, 03.65.Db, 03.65.Ge Nous etudions le probleme d'une particule quantique non relativiste sur la ligne reelle avec une interaction de portee nulle singuliere, idealisee et localisee a x = 0 (c.-a-d., une interaction ponctuelle en ce point). On peut deecrire ce type de systeme de deux facons. (i) On peut etudier le systeme alternatif libre en excluant le point x = 0 (sans le potentiel singulier), auquel cas, le point d'interaction est exclusivement encode dans la condition limite. (ii) On peut considerer l'interaction singuliere comme etant un potentiel local singulier. Nous analysons, relions et comparons ici ces deux approches equivalentes de facon pedagogique simple. Notre objectif est de presenter de facon accessible les points essentiels de ce problemes aux etudiants en fin de premier cycle. [Traduit par la Redaction], 1. Introduction Many aspects of point interactions (sharply localized singular perturbations or simply called contact interactions) such as the common Dirac delta function (or rather distribution), are fascinating, such as [...]

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam ([beta]=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied. [DOI: 10.1115/1.3025825] Keywords: nonhomogeneous, nonlinear, spring-mass system, rotating beam, multiple time scale

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the yon Karman equations with respect to displacements. The combination of the Rayleigh-Ritz method and the R-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method. [DOI: 10.1115/1.4001496] Keywords: circular plates with cut-outs, R-function method, conjugate modes, multiple scale method, internal resonances

This paper describes extensions and tests of characteristic methods for outlet boundary conditions in compressible solvers. Three methods based on the specification of incoming waves using one- and multidimensional approximations are extended to unstructured grids. They are first compared for weak to strong vortices propagating on low-to high-speed mean flows through outlet sections. A major issue is to determine the Mach number to be used in the specification of the transverse terms that must be taken into account in the incoming wave amplitude specifications. For the vortex computations, results show that the averaged Mach number leads to better results than its local value. The boundary conditions are then tested in a more complex case: the flow around a turbine blade. A reference solution using a long distance between the blade trailing edge and the outlet plane is first computed. For this solution, outlet boundary conditions have almost no effect on the flow around the blade: The distance between the trailing edge and the outlet plane is then shortened and the various characteristic treatments are compared, in which intense vortices cross the outlet plane. Results confirm the conclusions obtained on the simple vortex test case. DOI: 10.2514/1.J050391

A junction and drop-shaft boundary conditions (BCs) for one-dimensional modeling of transient flows in single-phase conditions (pure liquid) are formulated, implemented and their accuracy are evaluated using two computational fluid dynamics (CFD) models. The BCs are formulated in the case when mixed flows are simulated using two sets of governing equations, the Saint-Venant equations for the free-surface regions and the compressible water hammer equations for the pressurized regions. The proposed BCs handle all possible flow regimes and their combinations. The flow in each pipe can range from free surface to pressurized flow and the water depth at the junction or drop shaft can take on all possible levels. The BCs are applied to the following three cases: (1) a three-way merging flow; (2) a three-way dividing flow; and (3) a drop shaft connected to a single-horizontal pipe subjected to a rapid variation of the water surface level in the drop shaft. The flow regime for the first two cases range from free surface to pressurized flows, while for the third case, the flow regime is pure pressurized flow. For the third case, laboratory results as well as CFD results were used for evaluating its accuracy. The results suggest that the junction and drop-shaft BCs can be used for modeling transient free-surface, pressurized, and mixed flow conditions with good accuracy. DOI: 10.1061/(ASCE)HY.1943-7900.0000240 CE Database subject headings: Boundaries; Open channel flow; Pressurized flow; Sewers; Transient flow; Unsteady flow; Computational fluid dynamics technique. Author keywords: Boundary conditions; Mixed flow; Open-channel flow; Pressurized flow; Sewer; Transient flow; Unsteady flow; Computational fluid dynamics.

Elastic and elastic-plastic analyses of tensioned panels with an elliptical center notch are performed using the integrated local Petrov--Galerkin sinc method. The method is extended for analysis of boundary-value problems for nonlinear materials. The integrated local Petrov--Galerkin sinc method is used to obtain stress concentrations and stress intensities, using the J integral for both elastic and elastic-plastic plane-stress panels. The material is assumed to behave in a bilinear manner with kinematic hardening. The results are compared with the analytic solution for an infinite elastic panel in tension and with a finite-element solution for the panel with material nonlinearity. Good correlation is seen with stress concentration; although, the method tends to overestimate the stress concentrations. The nonlinear results compare well with the finite-element results. The accuracy deteriorates as the plastic zone propagates into areas less densely populated by the sinc points. For the elastic--plastic material, the J integral was estimated within 5% of the finite-element result, using the integrated local Petrov--Galerkin sinc method. DOI: 10.2514/1.J050281

Rotor aerodynamics is governed by wake geometry and strength. However, rotor wake characteristics computed by the rotor computational fluid dynamics are not clearly described due to numerical dissipation. To overcome this numerical problem, free-wake is used for wake simulation. The present free-wake describes the inboard vortices as well as the tip vortices of the blade. At each time step, the free-wake provides inflow and outflow conditions of the boundary of the Eulerian domain, and the Euler solver is used for solving the flow field near the rotor blade. Finally, the coupled method is compared with the conventional method and the experimental results. [DOI: 10.1115/1.4002110]

The interslice force function f(x) is a major assumption of the limit equilibrium method, which is important but has not been adequately considered in the past. In this paper, f(x) is taken as the control variable, and the upper and lower limits of the factor of safety for a general slope will be determined by a global optimization analysis. Based on this approach, f(x) will be determined and investigated. We demonstrate that f(x) cannot be arbitrarily assigned if a set of acceptable internal forces is required. The present approach can be presented practically as a lower bound approach with the advantage that failure to converge is virtually eliminated, which is not possible with all other existing 'rigorous' methods. The 'present proposal' attempts to answer several important questions in the basic theory of slope stability analysis, and provides a f(x) based on the lower bound approach statically admissible forces throughout the whole failure zone. Currently, different assumptions will give different factors of safety to the same problem, and this situation will be overcome by the use of the present proposal. The present proposal is also proven to give a result equal to the slip line solution for a simple footing on clay which is not possible for other classical slope stability methods, which has demonstrated that the applicability of the 'present proposal' for general difficult problems. DOI: 10.1061/(ASCE)GT.1943-5606.0000317 CE Database subject headings: Slope stability; Safety; Convergence; Limit equilibrium. Author keywords- Slope stability analysis; Factor of safety; Interslice force function; Global maximum; Lower bound theorem; Convergence.

In computational mechanics, strain softening generates ill-posed boundary value problems, which cannot be solved without being regularized, e.g., through the introduction of an internal length scale. This paper investigates how the internal length scale that is introduced by regularization may interact with the external length scale arising from boundary conditions in the particular case of a strain-softening bar of varying cross section and a nonlocal averaging regularization. The interaction of internal and external length scales is examined using an analytical closed-form solution for overnonlocal softening plasticity that derives from a Fredholm equation of the second kind. In the absence of external length (bars of constant section), the analysis shows that the overnonlocal averaging confines and smoothly distributes plastic strain into a localized band. The localization width, plastic strain distribution inside the band, and load-displacement response are controlled by the internal length of the averaging function and the overnonlocal weighting factor. In the presence of external length (bar of varying section), the analytical solution shows that the localization width is controlled by the interaction of external and internal length scales. This interaction is significant when the external and internal lengths are of comparable magnitude, and decreases when the external length becomes large compared to the internal length. The bandwidth is found to depend on the internal and external lengths and stress level while strain localizes, and to relate only to the internal length when the bar collapses. DOI: 10.1061/(ASCE)EM.1943-7889.0000145 CE Database subject headings: Strain; Localization; Plasticity; Bars. Author keywords: Strain localization; Nonlocal material; Strain softening plasticity; Regularization; Analytical solution.

We investigate the scattering process of plane waves by a conducting half-plane between two dielectric media by introducing novel boundary conditions, in terms of soft and hard surfaces. The cases of soft and hard half-planes are studied independently. The scattered waves are examined numerically. The numerical results show that the behavior of the fields is in harmony with the theory. The transition between the two dielectric media is continuous, and the structure of the method enables one also to examine more complex geometries, such as wedges having soft and hard boundary conditions. OCIS codes: 290.0290, 290.5825, 050.0050, 050.1960.

In the present work, the Jeffery--Hamel flow problem has been studied using both first-and second-order velocity-slip models, and then compared with the no-slip model. The objectives are to observe the behavior of the flow predicted by the two slip models and to establish criteria for using the two velocity-slip models. The study concentrates on examining the effect of the change in the Knudsen number (Kn) on the velocity profiles, magnitude of slip at the wall, and skin friction coefficient. Assuming that a difference between the two slip models of the order of 10% or less justifies the use of the simple first-order model, the transitional Kn numbers have been found. These Kn numbers depend on the flow direction, being either inflow or outflow. Also, there are three distinct regions that specify where to use each of the no-slip, first-order, and second-order slip models. Further, the reversal of the flow has been investigated as a function of the Kn number and for different Re. a, where Re is Reynolds number and a is the wall angle. Using the second-order slip models, it is found that as the Kn number increases, reversal occurs at Re. a smaller than the 10.31 value at which flow reversal happens in the no-slip model, and increasing the Kn number leads to a reduction in the skin friction coefficient in all cases except when reversal occurs. [DOI: 10.1115/1.4000918]

Vertical vibration of a rigid circular disc attached to the surface of a transversely isotropic half-space is considered in such a way that the axis of material symmetry is normal to the surface of the half-space and parallel to the vibration direction. By using Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations termed dual integral equations in the literature, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration the governing integral equation is numerically solved in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure, and the static impedance/compliance function are explicitly solved. The dynamic contact pressure under the disc and the impedance function are numerically evaluated, and it is shown that the singularity that exists at the edge of the disc is the same as the one obtained for the static case. In addition, the impedance functions evaluated here are identical to the solution given by Luco and Mita for the isotropic domain. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared. DOI: 10.1061/(ASCE)EM. 1943-7889.0000114 CE Database subject headings: Isotropy; Half space; Wave propagation; Integral equations; Soil-structure-interactions; Vibration. Author keywords: Rigid disc; Transversely isotropic; Half-space; Wave propagation; Dual integral equations; Impedance function; Compliance function; Soil-structure-interaction.

The oscillatory behavior of the transmission coefficient T as a function of energy is examined for an attractive square well and a rectangular barrier. We calculate T using resonant state boundary conditions and demonstrate that the maxima in T are correlated with the broad resonances generated by these potentials. For barrier potentials the maxima signify resonances occurring at energies above the barrier height. It is shown that the resonance position and width can also be generated from the complex poles of the amplitude of the transmitted plane wave. We also explain the relation between the positions of the resonances generated by the square well and the rectangular barrier to the energy eigenvalues of the corresponding rigid box with the same range. We show for a potential with an attractive well and a repulsive barrier that T exhibits oscillations when the particle energy is below the barrier, implying that in many cases the simple WKB type barrier penetration expression for T is not adequate. These features of T are likely to hold for most attractive potentials and flat repulsive barriers. We also discuss the attractive modified Poschl--Teller type potential for which T does not show oscillations as a function of energy. [c] 2010 American Association of Physics Teachers. [DOI: 10.1119/1.3276053]

Contact detection in cartilage contact mechanics is an important feature of any analytical or computational modeling investigation when the biphasic nature of cartilage and the corresponding tribology are taken into account. The fluid flow boundary conditions will change based on whether the surface is in contact or not, Which will affect the interstitial fluid pressurization. This in turn will increase or decrease the load sustained by the fluid phase, with a direct effect on friction, wear, and lubrication. In laboratory experiments or clinical hemiarthroplasty, when a rigid indenter or metallic prosthesis is used to apply load to the cartilage, there will not be any fluid flow normal to the surface in the contact region due to the impermeable nature of the indenter/prosthesis. In the natural joint, on the other hand, where two cartilage surfaces interact, flow will depend on the pressure difference across the interface. Furthermore, in both these cases, the fluid would flow freely in non-contacting regions. However, it should be pointed out that the contact area is generally unknown in advance in both cases and can only be determined as part of the solution. In the present finite element study, a general and robust algorithm was proposed to decide nodes in contact on the cartilage surface and, accordingly, impose the fluid flow boundary conditions. The algorithm was first tested for a rigid indenter against cartilage model. The algorithm worked well for two-dimensional four-noded and eight-noded axisymmetric element models as well as three-dimensional models. It was then extended to include two cartilages in contact. The results were in excellent agreement with the previous studies reported in the literature. [DOI: 10.1115/l.4000869] Keywords: articular cartilage, biphasic, contact mechanics, finite element, contact dependent flow

A discontinuous Galerkin method is applied to unstructured grids to simulate aeroacoustic propagation, modeled by the linearized Euler equations. On triangular and tetrahedral elements with straight edges, the quadrature-free form of the discontinuous Galerkin method is used. In addition to the classical linear treatment of wall boundaries, two treatments involving a second-order representation of the geometry are presented. The simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order, and demonstrates how the boundary treatment involving curved elements overcomes this restriction. The benefits of higher-order treatments are also assessed for more realistic geometries, namely a high-lift airfoil and an elliptical muffler. DOI: 10.2514/1.45353

We study apparent elastic moduli of trabecular bone, which is represented, for simplicity, by a two- or three-dimensional periodic cellular network. The term 'apparent' refers to the case when the region used in calculations (or specimen size) is smaller than a representative volume element and the moduli depend on the size of that region and boundary conditions. Both the bone tissue forming the network and the pores (represented by a very soft material) are assumed, for simplicity, as homogeneous, linear elastic, and isotropic. In order to investigate the effects of scale and boundary conditions on the moduli of these networks we vary. the specimen size and apply four different boundary, conditions: displacement, traction, mixed, and periodic. The analysis using periodic boundary conditions gives the effective moduli, while the displacement, traction, and mixed boundary conditions give apparent moduli. The apparent moduli calculated using displacement and traction boundary, conditions bound the effective moduli from above and below, respectively. The larger is the size of the region used in our calculations, the closer are the bounds. Our choice of mixed boundary conditions gives results that are very close to those obtained using periodic boundary conditions. We conduct this analysis computationally using a finite element method. We also investigate the effect of mismatch in elastic moduli of bone tissue and soft fill, trabecular bone structure geometry, and bone tissue volume fraction on the apparent elastic moduli of idealized periodic models of trabecular bone. This study gives guidance on how the size of the specimen and boundary conditions (used in experiments or simulations) influence elastic moduli of cellular materials. This approach is applicable to heterogeneous materials in general. [DOI: 10.1115/1.4000192] Keywords: apparent elastic moduli, trabecular bone, scale and boundary conditions effects, representative volume element, periodic microstructure

Proximity effect systems in superconducting films can be modeled by a one-dimensional Schrodinger equation. Several systems are studied using Dirichlet and Neumann boundary conditions. It is observed that the two boundary conditions have a dramatic effect on the lowest eigenstate allowed in these systems, and this points to unusual behavior for solutions of Schrodinger's equation in certain potential wells and proximity effect systems. [DOI: 10.1119/1.3078414]

The class of uniaxial electromagnetic IB media involving five medium parameters is defined as a special case of the previously defined class of general IB media (or skewon-axion media) involving 16 parameters. The problem of plane-wave reflection from and transmission through the interface of a uniaxial IB-medium half space is analyzed. It is shown that, for general values of the medium parameters, waves polarized TE and TM with respect to the normal direction are reflected as from the respective PEC and PMC boundaries. Unlike the anisotropic soft-and-hard surface which has a somewhat similar property, the uniaxial IB interface is isotropic in the plane of the interface. As a consequence, a novel class of electromagnetic boundaries can be defined for general fields requiring vanishing of the normal components of both D and B vectors at the boundary. Another realization of such a boundary in terms of an anisotropic metamaterial with zero axial permittivity and permeability is discussed. Index Terms--Boundary conditions, electromagnetic theory, metamaterials.

The problem of thermoelasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a boundary value problem of one dimensional semi-infinite medium heated [...]

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials. [DOI: 10.1115/1.2936241]

Rigorous scale-dependent bounds on the constitutive response of random polycrystalline aggregates are obtained by setting up two stochastic boundary value problems (Dirichlet and Neumann type) consistent with the Hill condition. This methodology enables one to estimate the size of the representative volume element (RVE), the cornerstone of the separation of scales in continuum mechanics. The method is illustrated on the single-phase and multiphase aggregates, and, generally, it turns out that the RVE is attained with about eight crystals in a 3D system. From a thermodynamic perspective, one can also estimate the scale dependencies of the dissipation potential in the velocity space and its complementary potential in the force space. The viscoplastic material, being a purely dissipative material is ideally suited for this purpose. [DOI: 10.1115/1.2912999] Keywords: homogenization, random polycrystals, representative volume element (RVE), bounds, plasticity

A new numeric-analytic technique is developed for two-point nonlinear boundary-value problems (BVPs) of engineering interest. The analytic part of the method is based on a conventional Adomian decomposition method (ADM). However, given a discretization of the one-dimensional domain, the present algorithm applies the ADM, repetitively over successive intervals and exploits a shooting algorithm to solve the BVPs. Apart from a very high rate of convergence as the discretization is made finer, yet another significant advantage of the method is that it provides the solution in a piecewise functional form and one can finally arrive at a continuous form of the global solution. The procedure is used to study planar, large-deflection (Elastica) problem of a cantilever beam subjected to a transverse, concentrated load, at its free end. Moreover the elastoplastic behavior of a cantilever is also studied. Comparisons with exact solutions as well as with results via a few other competing algorithms demonstrate the remarkable accuracy of the proposed method. DOI: 10.1061/(ASCE)0733-9399(2007)133:10(1124) CE Database subject headings: Decomposition; Numerical analysis; Algorithms; Differential equations.

The implementation and evaluation of a spherical perfectly matched layer (PML) within a Cartesian finite element method context using standard curl-conforming elements is presented in this paper. Results are compared to the long-standing 1st order absorbing boundary condition (ABC) and a new, rigorous implementation of a 2nd order ABC for curl-conforming elements. The 4 and 8 layer spherical PMLs are shown to offer very attractive levels of absorption, with reflections on the order of -60 to -70dB demonstrated. Numerical tests show that the guidelines for Cartesian PML absorbers, in terms of maximum conductivity, also carry over to the spherical PML. The 2nd order ABC is also shown to offer very good performance. Finally, coding issues for both the spherical PML and the analytical ABCs are briefly addressed. Index Terms--Absorbing boundary conditions (ABC), finite element method (FEM), perfectly matched layer (PML).

The Green's functions for the mixed potential electric field integral equation are derived for a rectangular waveguide with dielectric-filled corrugations supporting left- as well as right-hand propagation. Under the asymptotic boundary conditions assumption, the expressions of the Green's dyadic components can be decomposed into terms that represent conventional waveguide modes and others that represent hybrid metaguide modes. A simple approach is used to find the poles of the spectral domain expressions before obtaining the spatial domain expressions using the inverse Fourier transform. The dispersion diagram reflects very interesting characteristics for the structure and the deviation from the conventional case. The derived Green's function is verified by considering the problem of a probe excitation and comparing the input impedance obtained using a moment method procedure based on the present theory and the finite-difference time-domain method. Index Terms--Corrugated waveguide, green's function, lefthand propagation, metamaterial transmission line.

Numerical method for modeling the E-polarized wave scattering by electrically large quasioptical two-dimensional (2-D) reflectors is presented. Reflectors are assumed zero-thickness and perfectly electrically conducting. Efficient numerical solution is obtained from the coupled singular integral equations discretized using new quadrature formulas of interpolation type. It has controlled accuracy and deals with small-size matrices. To simulate a small-horn feeding, the incident field is taken as a beam generated by a complex-source-point (CSP) current. Presented numerical results validate empirical rule of -10 dB edge illumination needed to provide the best electromagnetic performance of reflector. Index Terms--Boundary integral equations, method of discrete singularities (MDS), reflector antennas, scattering.

Abstract: We consider boundary value problems of anti-plane shear in the static theory of linear piezoelectricity. Using the boundary integral equation method, we reduce the problems to systems of singular [...]

Impedance boundary is generally considered as an approximate model for material interfaces. Considering an electromagnetic field in a certain wave-guiding anisotropic material it is shown that a slab of such a material backed by a perfect electric conductor (PEC) plane can be exactly represented by impedance-boundary conditions. As a result of the analysis, a novel explicit relation is derived between the surface admittance dyadic of the impedance boundary and the material and geometric parameters of the anisotropic slab which can be used to realize a given admittance dyadic. The relation is verified with results known for the perfect electromagnetic (PEMC) boundary and bi-axial material, considered as special cases of the theory. Index Terms--Boundary conditions, electromagnetics, impedance boundary.

In this chapter, we investigate the operator l[y] = [y.sup.(n)] + [p.sub.2](x)[y.sup.(n-2)] + ... + [p.sub.n](x)y, x [member of] [0, 1], [p.sub.k](x) [member of] C[0, 1], (5.1) with arbitrary decomposing [...]