Trapped modes in thin and infinite ladder like domains. Part 1 : existence results

International audienceThe present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $\epsilon > 0$) whose limit (as $\epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $\epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that ε is small enough. Numerical experiments illustrate the theoretical results

Guides d'ondes ouverts : théorie et calculs numériques.

The present work deals with propagation of acoustic waves in periodic media. These media have particularly interesting properties since the spectrum associated with the underlying wave operator in such media has a band-gap structure: there exist intervals of frequences for which monochromatic waves do not propagate. Moreover, by introducing linear defects in this kind of media, one can create guided modes inside the bands of forbidden frequences. In this work we show that it is possible to create such guided modes in the case of particular periodic media of grid type: more precisely, the periodic domain in question is R2 minus an infinite set of rectangular obstacles periodically spaced in two orthogonal directions (the distance between two neighbour obstacles being ε), which is locally perturbed by diminishing the distance between two columns of obstacles. The results are extended to the 3D case.This work has a theoretical and a numerical aspect. From the theoretical point of view the analysis is based on the fact that, ε being small, the spectrum of the operator associated with our problem is "close" to the spectrum of a problem posed on a graph which is a geometric limit of the domain as ε tends to 0. However, for the limit graph the spectrum can be computed explicitly. Then, we study the spectrum of the non-limit operator using asymptotic analysis. Theoretical results are illustrated by numerical computations obtained with a numerical method developed for study of periodic media: this method is based on the reduction of the initial (linear) eigenvalue problem posed in an unbounded domain to a non-linear problem posed in a bounded domain (using the exact Dirichlet- to-Neumann operator).Cette thèse porte sur la propagation des ondes acoustiques dans des milieux périodiques. Ces milieux ont des propriétés remarquables car le spectre associée à l’opérateur d’ondes dans ces milieux a une structure de bandes : il existe des plages de fréquences dans lesquelles les ondes monochromatiques ne se propagent pas. Plus intéressant encore, en introduisant des défauts linéiques dans ce type de milieux, on peut créer des modes guidés à l’intérieur de ces bandes de fréquences interdites. Dans ce manuscrit nous montrons qu’il est possible de créer de tels modes guidés dans le cas de milieux périodiques particuliers de type quadrillage : plus précisément, le domaine périodique considéré est constitué du plan R2 privé d’un ensemble infini d’obstacles rectangulaires régulièrement espacés (d’une distance ε) dans deux directions orthogonales du plan, que l’on perturbe localement en diminuant la distance entre deux colonnes d’obstacles. Les résultats sont ensuite étendus au cas 3D.Ce travail comporte un aspect théorique et un aspect numérique. Du point de vue théo- rique l’analyse repose sur le fait que, comme ε est petit, le spectre de l’opérateur associé à notre problème est "proche" du spectre d’un problème posé sur le graphe obtenu comme la limite géométrique du domaine quand ε tend vers 0. Or, pour le graphe limite, il est possible de calculer explicitement le spectre. Ensuite, en utilisant des méthodes d’analyse asymptotique on étudie le spectre de l’opérateur non-limite. On illustre les résultats théo- riques par des résultats numériques obtenus à l’aide d’une méthode numérique spéciale- ment dédiée aux milieux périodiques : cette dernière est basée sur la réduction du problème de valeurs propres initial (linéaire) posé dans un domaine non-borné à un problème non- linéaire posé dans un domaine borné (en utilisant l’opérateur de Dirichlet-to-Neumann exact)

Guides d'ondes périodiques ouverts : Théorie et calcul

The present work deals with propagation of acoustic waves in periodic media. Thesemedia have particularly interesting properties since the spectrum associated with theunderlying wave operator in such media has a band-gap structure: there exist intervals offrequences for which monochromatic waves do not propagate. Moreover, by introducinglinear defects in this kind of media, one can create guided modes inside the bands offorbidden frequences. In this work we show that it is possible to create such guidedmodes in the case of particular periodic media of grid type: more precisely, the periodicdomain in question is R2 minus an infinite set of rectangular obstacles periodically spacedin two orthogonal directions (the distance between two neighbour obstacles being "),which is locally perturbed by diminishing the distance between two columns of obstacles.The results are extended to the 3D case.This work has a theoretical and a numerical aspect. From the theoretical point of view theanalysis is based on the fact that, " being small, the spectrum of the operator associatedwith our problem is "close" to the spectrum of a problem posed on a graph which is ageometric limit of the domain as " tends to 0. However, for the limit graph the spectrumcan be computed explicitly. Then, we study the spectrum of the non-limit operatorusing asymptotic analysis. Theoretical results are illustrated by numerical computationsobtained with a numerical method developed for study of periodic media: this method isbased on the reduction of the initial (linear) eigenvalue problem posed in an unboundeddomain to a non-linear problem posed in a bounded domain (using the exact Dirichletto-Neumann operator).Cette thèse porte sur la propagation des ondes acoustiques dans des milieux périodiques.Ces milieux ont des propriétés remarquables car le spectre associée à l’opérateur d’ondesdans ces milieux a une structure de bandes : il existe des plages de fréquences danslesquelles les ondes monochromatiques ne se propagent pas. Plus intéressant encore, enintroduisant des défauts linéiques dans ce type de milieux, on peut créer des modes guidésà l’intérieur de ces bandes de fréquences interdites. Dans ce manuscrit nous montrons qu’ilest possible de créer de tels modes guidés dans le cas de milieux périodiques particuliersde type quadrillage : plus précisément, le domaine périodique considéré est constitué duplan R2 privé d’un ensemble infini d’obstacles rectangulaires régulièrement espacés (d’unedistance ") dans deux directions orthogonales du plan, que l’on perturbe localement endiminuant la distance entre deux colonnes d’obstacles. Les résultats sont ensuite étendusau cas 3D.Ce travail comporte un aspect théorique et un aspect numérique. Du point de vue théoriquel’analyse repose sur le fait que, comme " est petit, le spectre de l’opérateur associé ànotre problème est "proche" du spectre d’un problème posé sur le graphe obtenu commela limite géométrique du domaine quand " tend vers 0. Or, pour le graphe limite, il estpossible de calculer explicitement le spectre. Ensuite, en utilisant des méthodes d’analyseasymptotique on étudie le spectre de l’opérateur non-limite. On illustre les résultats théoriquespar des résultats numériques obtenus à l’aide d’une méthode numérique spécialementdédiée aux milieux périodiques : cette dernière est basée sur la réduction du problèmede valeurs propres initial (linéaire) posé dans un domaine non-borné à un problème nonlinéaireposé dans un domaine borné (en utilisant l’opérateur de Dirichlet-to-Neumannexact)

Existence of guided waves due to a lineic perturbation of a 3D periodic medium

International audienceIn this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order ε > 0, is supposed to be small. We prove that, for ε small enough, shrinking the section of one line of the grating by a factor of √ µ (0 < µ < 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ε) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as ε tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly

Trapped modes in thin and infinite ladder like domains. Part 2 : asymptotic analysis and numerical application

International audienceWe are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter ε > 0) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results

Trapped modes in thin and infinite ladder like domains: existence and asymptotic analysis

We are interested in a 2D propagation medium which is a localized perturbation of a reference homogeneous periodic medium. This reference medium is a "thick graph", namely a thin structure (the thinness being characterized by a small parameter $\varepsilon$ > 0) whose limit (when $\varepsilon$ tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. The question we investigate is whether such a geometrical perturbation is able to produce localized eigenmodes. We have investigated this question when the propagation model is the scalar Helmholtz equation with Neumann boundary conditions . This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We use a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough and construct an asymptotic expansion of the eigenvalues with respect to the small parameter. Following this approach, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung. Using the matched asymptotic expansion method, we obtain an asymptotic expansion at any order of the eigenvalues, which can be used for instance to compute a numerical approximations of these eigenvalues and associated eigenvectors

COMPARATIVE ESTIMATION OF BIOCHEMICAL COMPOSITION OF FRUITS OF ERICACEAE SPECIES UNDER CONDITIONS OF BELARUS

Thus, as a result of biochemical screening of 30 taxons of Ericaceae species taking into account 32 parameters in a long-term cycle of observation it has been established the different degree of dependence of variability level of biochemical structure components of fruits of alien crops on genotype and a hydrothermal mode of the period of their maturing has been revealed. The similarity of parameters of genotypic variability of some traits of all investigated Ericaceae species is also revealed: low one – for the general contents in fruits of soluble sugars, flavonols, of potassium, calcium, magnesium and high one – for the contents of anthocyanins, and also the ratio of fractions of pectinaceous substances and bioflavonoids.Specific features of genetic determinacy of the analyzed traits, testifying the greatest degree of its displays of V. corymbosum for total accumulation of soluble sugars and bioflavonoids, flavonols contents, calcium and magnesium and by the lowest degree – for the contents of titratable acids, vitamin C, anthocyanins and values of a sugar-acid index were established. If to mention V. vitis-idaea L., the parameters of general accumulation in fruits of soluble sugars, dry substances and all major mineral elements were characterized by the greatest degree of genetic determinacy, whereas by the least degree – the contents of anthocyanins, catechines and tannins. If to mention V. macrocarpon, the parameters of accumulation in fruits of dry substances, nitrogen, potassium, calcium, phenol-carboxylic acids have been noted by the most expressed genetic determinacy, and by the least expressed – the contents of anthocyanins, sucrose and pectinaceous substances in fruits