This paper deals with generalization fo Gershgorin's theorem. This theorem is investigated and generalized in terms of contour integrals, directed graphs, convex analysis, and clock matrices. These results are shown to apply to some specified matrices such as stable and stochastic matrices and some examples will show the relationship of eigenvalue inclusion regions among them.

With increasing complexity of microwave integrated circuits and tendency towards building integrated modules, real estate in printed circuit boards becomes more at premium. On the other hand, building MIC's on a single semiconductor substrate such as GaAs has other drawbacks as substrate requirements for different components are sometimes contradictory. This has motivated researchers to consider multi-layer and stacked designs. Multi-layer planar circuits offer advantages that cannot be equaled by traditional single layer designs. In this respect, a new class of planar structures, based upon a multi-layered stack of dual-mode stripline or microstrip patches is becoming increasingly popular. In the new stacked design coupling between planar circuits separated by a ground plane is accomplished through coupling apertures in the common ground plane. <br ><br /> This thesis is about developing a new approximate multiport network model for fast analysis of multi-layered planar structures with ground plane slots. To extend applicability of multiport network model (MNM) to the class of planar structures containing ground plane slots, a generalized network formulation for aperture problems is combined with traditional MNM to account for the presence of the slot. To this end, the slot is replaced by an unknown equivalent surface magnetic current. Slot ports are defined in terms of electric and magnetic fields over the slot in accordance with the generalized network formulation for aperture problems. While traditional MNM for planar circuits is based on generalized impedance matrices, we adopt a hybrid matrix approach for multi-layer structures. The hybrid matrix consists of four sub-matrices that relate terminal voltages and currents of edge and slot ports. The same generalized impedance matrix in the absence of the slot can be used to relate terminal voltages and currents of edge ports when the slot ports are short-circuited. Open circuit voltage at edge ports due to terminal voltages at slot ports and terminal currents at slot ports due to input currents at edge ports are represented by two transfer matrices. Both these transfer matrices can be calculated from 2D analysis which only considers <em>TM^z</em> modes. <br ><br /> Interaction among slot ports, represented by a generalized admittance matrix, however, requires considering both <em>TM^z</em> and <em>TE^z</em> modes. This generalized admittance matrix is obtained from tangential component of the magnetic field over the slot due to the equivalent surface magnetic current and relates terminal voltages and currents of slot ports. Full modal expansion consisting of both <em>TM^z</em> and <em>TE^z</em> modes is used to compute the generalized admittance matrix of a slot in a regularly shaped planar cavity. For irregularly shaped patches, modal expansion is not available. Instead, a new contour integral equation for magnetic field, derived for the first time in this thesis, is combined with complex images method for calculation of generalized admittance matrix of a slot radiating in a planar cavity of arbitrary shape. <br ><br /> Once the hybrid matrix representation of a planar circuit on a ground plane containing a slot is derived, it can be connected to the hybrid matrix of any other planar circuit on the other side of the ground plane. This can be done by enforcing network equivalent of continuity of tangential fields across the slot. This leads to a generalized impedance matrix for the multi-layer structure relating terminal voltages and currents of edge ports of both planar circuits. <br ><br /> To show the accuracy of the proposed method of analysis, several proof-of-concept structures have been analyzed by both this method and ANSOFT HFSS full-wave simulator as a reference. In most cases excellent agreement is achieved in predicting the return loss and radiation patterns of these multi-layer structures which proves the validity of the proposed approach for fast analysis and design of multi-layer planar structures.

With increasing complexity of microwave integrated circuits and tendency towards building integrated modules, real estate in printed circuit boards becomes more at premium. On the other hand, building MIC's on a single semiconductor substrate such as GaAs has other drawbacks as substrate requirements for different components are sometimes contradictory. This has motivated researchers to consider multi-layer and stacked designs. Multi-layer planar circuits offer advantages that cannot be equaled by traditional single layer designs. In this respect, a new class of planar structures, based upon a multi-layered stack of dual-mode stripline or microstrip patches is becoming increasingly popular. In the new stacked design coupling between planar circuits separated by a ground plane is accomplished through coupling apertures in the common ground plane. <br ><br /> This thesis is about developing a new approximate multiport network model for fast analysis of multi-layered planar structures with ground plane slots. To extend applicability of multiport network model (MNM) to the class of planar structures containing ground plane slots, a generalized network formulation for aperture problems is combined with traditional MNM to account for the presence of the slot. To this end, the slot is replaced by an unknown equivalent surface magnetic current. Slot ports are defined in terms of electric and magnetic fields over the slot in accordance with the generalized network formulation for aperture problems. While traditional MNM for planar circuits is based on generalized impedance matrices, we adopt a hybrid matrix approach for multi-layer structures. The hybrid matrix consists of four sub-matrices that relate terminal voltages and currents of edge and slot ports. The same generalized impedance matrix in the absence of the slot can be used to relate terminal voltages and currents of edge ports when the slot ports are short-circuited. Open circuit voltage at edge ports due to terminal voltages at slot ports and terminal currents at slot ports due to input currents at edge ports are represented by two transfer matrices. Both these transfer matrices can be calculated from 2D analysis which only considers <em>TM^z</em> modes. <br ><br /> Interaction among slot ports, represented by a generalized admittance matrix, however, requires considering both <em>TM^z</em> and <em>TE^z</em> modes. This generalized admittance matrix is obtained from tangential component of the magnetic field over the slot due to the equivalent surface magnetic current and relates terminal voltages and currents of slot ports. Full modal expansion consisting of both <em>TM^z</em> and <em>TE^z</em> modes is used to compute the generalized admittance matrix of a slot in a regularly shaped planar cavity. For irregularly shaped patches, modal expansion is not available. Instead, a new contour integral equation for magnetic field, derived for the first time in this thesis, is combined with complex images method for calculation of generalized admittance matrix of a slot radiating in a planar cavity of arbitrary shape. <br ><br /> Once the hybrid matrix representation of a planar circuit on a ground plane containing a slot is derived, it can be connected to the hybrid matrix of any other planar circuit on the other side of the ground plane. This can be done by enforcing network equivalent of continuity of tangential fields across the slot. This leads to a generalized impedance matrix for the multi-layer structure relating terminal voltages and currents of edge ports of both planar circuits. <br ><br /> To show the accuracy of the proposed method of analysis, several proof-of-concept structures have been analyzed by both this method and ANSOFT HFSS full-wave simulator as a reference. In most cases excellent agreement is achieved in predicting the return loss and radiation patterns of these multi-layer structures which proves the validity of the proposed approach for fast analysis and design of multi-layer planar structures.

Cette étude vise à i) déterminer le trajet de fissuration et la vitesse de propagation de fissure de fatigue dans un joint soudé sollicité en mode mixte ii) de proposer une analyse des interactions fissuration par fatigue/microstructure pour deux classes d'alliages soudés et iii) de proposer des critères de propagation de fissure, basé sur l'analyse des facteurs d'intensité de contrainte (FIC) pour les trois modes de fissuration.Les essais ont été réalisés pour un acier ferritique inoxydable, K41X soudé par gaz inerte, sur des éprouvettes à entaille centrale sous sollicitation uniaxiale pour différentes orientations d'entaille. Pour un assemblage bi-matériaux d'alliages de titane, obtenu par soudure laser de Ti17 et de Ti6242, les essais ont été effectués en condition biaxiale coplanaire sur une éprouvette en croix à entaille centrale à différents taux de cisaillement macroscopique.Pour l'acier, la fissure se propage en mode mixte trans- et intergranulaire dans le métal de base, et en mode transgranulaire uniquement dans la zone fondue (ZF). Néanmoins, c'est seulement en présence de cisaillement macroscopique induit par la géométrie d'entaille que la soudure a conduit à une accélération de la vitesse de fissure par rapport au métal de base. Pour Ti17 le chemin de propagation de fissure et la vitesse de propagation sont très réguliers alors qu'ils sont extrêmement oscillants pour Ti6242, car corrélés aux aiguilles α. La vitesse de fissuration est plus faible dans Ti6242 que dans Ti17 pour une sollicitation d'équibitraction, alors qu'elle est plus élevée dans Ti6242 que dans Ti17 pour un cisaillement macroscopique. Pour les éprouvettes soudées Ti17-Ti6242, le chemin de propagation est relativement régulier dans la ZF mais peut être piloté par l'interface zone affectée thermiquement (ZAT)/ZF. La vitesse est plus élevée dans la ZF que pour les deux métaux de base, alors qu'elle n'est que peu modifiée dans les ZAT.La régularité de la propagation de fissure dans Ti17 a permis d'utiliser ce cas comme un cas de référence dans le cadre de la mécanique linéaire de la rupture et ainsi proposer un critère de ∆Keq fonction des modes de sollicitations I, II et III. Ces résultats ont été obtenus en modélisant le chemin de fissure en surface et le déversement de fissure. L'analyse comparative des propriétés des différents matériaux testés a été menée dans ce cadre. Une analyse de sensibilité à la précision de la modélisation tridimensionnelle de la géométrie de fissure met en évidence le rôle prépondérant de celle-ci sur l'estimation des FIC en mode III et par conséquent sur la valeur de ∆Keq. Cette analyse permet d'expliquer les oscillations de vitesse constatées pour l'alliage Ti6242. / This study is devoted to i) the experimental characterization of fatigue crack path and fatigue crack growth rate (FCGR) for welded materials under mixed mode of loading ii) the analysis of fatigue crack to microstructure interactions for two types of welded materials and iii) the identification of a relevant FCGR criterion function of the mode mixity.The experimental characterization was achieved for a ferritic stainless steel, welded by metal inert gas, using CCT specimens for different orientations of the notch. Biaxial testing was achieved using central crack cross-shaped specimen, varying the shear to opening loading ratio, for a dissimilar welded joint obtained by laser welding of Ti17 and Ti6242 Ti base alloys.For the ferritic stainless steel, the crack is both trans- and intergranular for the base metal, whereas it becomes mainly transgranular for the welded specimen. Nevertheless, very slight modification of the FCGR, comparing base metal and welded material, is observed. The welding was seen to be detrimental only in the case of macroscopic shear loading implied by the geometry of the notch. For Ti17, crack path is very smooth and FCGR evolves regularly as a function of the stress intensity factor (SIF). Instead of what, for Ti6242 alloy, both the crack path and the FCGR present large amplitudes of oscillation, due to a strong interaction with alpha needles. The FCGR is lower in Ti6242 than in Ti17 for equibiaxial fatigue, whereas the FCGR is higher in Ti6242 than in Ti17 for macroscopic shear fatigue loading. For welded Ti17-Ti6242 specimens, FCGR is higher than observed in base metal for crack within the fusion zone (FZ), and tends to the FCGR of the associated base metal to each heat-affected zone (HAZ-Ti17 and HAZ-Ti6242).The case of Ti17 was seen to be relevant to determine an equivalent SIF function of mode mixity within the scope of LEFM. An original criterion has been established taking into consideration mode I, II and III. The numerical model describes explicitly both surface crack path and flat to slant orientation of the crack. This criterion has been successfully applied to both Ti and Fe base alloys, for base metal as well as for welded materials in order to determine the impact of welding on FCGR. At last but not least, the sensitivity of SIF values to the accuracy of the 3D modelling of the crack surface has been tested. Thus the local roughness of the crack path is seen to drastically impact the out-of-plane shear mode, which is in turn fully consistent with acceleration/deceleration of the crack observed for Ti6242 alloy.

O trabalho apresenta uma análise do desempenho de vários métodos de extração de fatores de intensidade de tensão a partir de soluções numéricas obtidas com o método dos elementos finitos generalizados (MEFG). A convergência dos fatores de intensidade de tensão é comparada com a da energia de deformação a fim de investigar a superconvergência dos métodos. Para extração dos fatores de intensidade de tensão e o cálculo da taxa de energia disponibilizada para propagação da fissura, implementam-se os métodos da integral de contorno (MIC), da função cutoff (MFC) e da integral-J no contexto do MEFG. Desenvolve-se a formulação dos métodos de extração de forma a obter uma implementação independente da malha utilizada na modelagem do problema. Aplica-se a extração dos fatores de intensidade de tensão, para modos puros e mistos, em problemas clássicos da mecânica da fratura. Verifica-se a convergência dos fatores de intensidade de tensão e da taxa de energia disponibilizada para a propagação da fissura, obtidos com cada método de extração, com o enriquecimento da ordem polinomial da solução do MEFG. Investiga-se a robustez dos métodos com relação ao tamanho dos domínios de extração / The performance of several techniques to extract stress intensity factors (SIF) from numerical solutions computed with the generalized finite element method (GFEM) is investigated. The convergence of the stress intensity factors is compared with the convergence of strain energy with the aim of investigate the superconvergence of the methods. The contour integral (CIM), the cutoff function (CFM) and the J-integral methods are considered to compute stress intensity factors and energy release rate. The proposed implementation of the extraction techniques is completely independent of the discretization used. Several numerical examples demonstrating the convergence of the computed stress intensity factors and the energy release rate, with the increasing of p order of the GFEM solution, are presented

O trabalho apresenta uma análise do desempenho de vários métodos de extração de fatores de intensidade de tensão a partir de soluções numéricas obtidas com o método dos elementos finitos generalizados (MEFG). A convergência dos fatores de intensidade de tensão é comparada com a da energia de deformação a fim de investigar a superconvergência dos métodos. Para extração dos fatores de intensidade de tensão e o cálculo da taxa de energia disponibilizada para propagação da fissura, implementam-se os métodos da integral de contorno (MIC), da função cutoff (MFC) e da integral-J no contexto do MEFG. Desenvolve-se a formulação dos métodos de extração de forma a obter uma implementação independente da malha utilizada na modelagem do problema. Aplica-se a extração dos fatores de intensidade de tensão, para modos puros e mistos, em problemas clássicos da mecânica da fratura. Verifica-se a convergência dos fatores de intensidade de tensão e da taxa de energia disponibilizada para a propagação da fissura, obtidos com cada método de extração, com o enriquecimento da ordem polinomial da solução do MEFG. Investiga-se a robustez dos métodos com relação ao tamanho dos domínios de extração / The performance of several techniques to extract stress intensity factors (SIF) from numerical solutions computed with the generalized finite element method (GFEM) is investigated. The convergence of the stress intensity factors is compared with the convergence of strain energy with the aim of investigate the superconvergence of the methods. The contour integral (CIM), the cutoff function (CFM) and the J-integral methods are considered to compute stress intensity factors and energy release rate. The proposed implementation of the extraction techniques is completely independent of the discretization used. Several numerical examples demonstrating the convergence of the computed stress intensity factors and the energy release rate, with the increasing of p order of the GFEM solution, are presented

This work deals with the method for analysis of one-port microstrip circuits which is called as contour-integral method. The general procedure for evaluating of important parameters concerning the analysis of microstrip antennas is given. Next, the geometrical parameters necessary for the calculation of elements in U and H matrices are numerically solved. With the help of these matrices, the input impedance or the reflection coefficient can be easily found.

Cette thèse porte sur certains aspects symboliques-numériques du problème des paires invariantes pour les polynômes de matrices. Les paires invariantes généralisent la définition de valeur propre / vecteur propre et correspondent à la notion de sous-espaces invariants pour le cas nonlinéaire. Elles trouvent leurs applications dans le calcul numérique de plusieurs valeurs propres d’un polynôme de matrices; elles présentent aussi un intérêt dans le contexte des systèmes différentiels. En utilisant une approche basée sur les intégrales de contour, nous déterminons des expressions du nombre de conditionnement et de l’erreur rétrograde pour le problème du calcul des paires invariantes. Ensuite, nous adaptons la méthode des moments de Sakurai-Sugiura au calcul des paires invariantes et nous étudions le comportement de la version scalaire et par blocs de la méthode en présence de valeurs propres multiples. Le résultats obtenus à l’aide des approches directes peuvent éventuellement être améliorés numériquement grâce à une méthode itérative: nous proposons ici une comparaison de deux variantes de la méthode de Newton appliquée aux paires invariantes. Le problème des solvants de matrices est très proche de celui des paires invariants. Le résultats présentés ci-dessus sont donc appliqués au cas des solvants pour obtenir des expressions du nombre de conditionnement et de l’erreur, et un algorithme de calcul basé sur la méthode des moments. De plus, nous étudions le lien entre le problème des solvants et la transformation des polynômes de matrices en forme triangulaire. / In this thesis, we study some symbolic-numeric aspects of the invariant pair problem for matrix polynomials. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. They have applications in the numeric computation of several eigenvalues of a matrix polynomial; they also present an interest in the context of differential systems. Here, a contour integral formulation is applied to compute condition numbers and backward errors for invariant pairs. We then adapt the Sakurai-Sugiura moment method to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues, and we analyze the behavior of the scalar and block versions of the method in presence of different multiplicity patterns. Results obtained via direct approaches may need to be refined numerically using an iterative method: here we study and compare two variants of Newton’s method applied to the invariant pair problem. The matrix solvent problem is closely related to invariant pairs. Therefore, we specialize our results on invariant pairs to the case of matrix solvents, thus obtaining formulations for the condition number and backward errors, and a moment-based computational approach. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

Tato práce se zabývá analýzou power-bus struktur využitím metody založené na formulaci problému 2D hraniční integrální rovnicí ve frekvenční oblasti. Uvedená metoda byla implementována v Matlabu. Program umožňuje analyzovat obecné polygonální power-bus struktury s možností nastavení parametrů substrátu a libovolného počtu a umístění budících portů. Výstupem programu je frekvenční závislost rozložení elektrického pole mezi deskami struktury, vztahy mezi porty struktury vyjádřené např. impedanční maticí a vyzařovací diagram. Dále byla implementována možnost převodu výsledné impedanční matice do tzv. Touchstone formátu, pomocí něhož je možné modelovat analyzované struktury jako obecné N-porty (např. v ANSOFT Designeru), což umožňuje analýzu power-bus struktur s dalšími obvodovými prvky. Výsledky byly ověřeny pomocí existujících analytických vztahů pro jednoduché obdélníkové struktury, využitím komerčního simulačního programu a praktickým experimentem.