The theory of electromagnetic characteristic modes was published roughly forty years ago, for both conducting and penetrable objects. However, while the characteristic mode analysis of conducting objects has found renewed interest as a tool for antenna designers, computed results for the characteristic mode eigenvalues, eigencurrents and eigenfields for penetrable objects have not appeared, not even in the seminal papers on the subject. In this thesis both volume and surface integral equation formulations are used to compute the characteristic modes of penetrable objects for what appears to be the first time. This opens the way for the use of characteristic mode theory in the design of antennas made of penetrable material whose polarization current densities constitute the main radiating mechanism of the antenna. Volume formulations are shown to be reliable but computationally burdensome. It is demonstrated that surface formulations are computationally more efficient, but obtrude some non-physical modes in addition to the physical ones. Fortunately, certain field orthogonality checklists can be used to provide a straightforward means of unambiguously selecting only the physical modes. The sub-structure characteristic mode concept is extended to problems involving both perfectly conducting and penetrable materials. It is also argued that sub-structure modes can be viewed as characteristic modes that implicitly use modified Green’s functions, but without such Green’s functions being needed explicitly. This makes the concept really practical, since the desired modified Green’s functions are not known explicitly in most cases.

The theory of electromagnetic characteristic modes was published roughly forty years ago, for both conducting and penetrable objects. However, while the characteristic mode analysis of conducting objects has found renewed interest as a tool for antenna designers, computed results for the characteristic mode eigenvalues, eigencurrents and eigenfields for penetrable objects have not appeared, not even in the seminal papers on the subject. In this thesis both volume and surface integral equation formulations are used to compute the characteristic modes of penetrable objects for what appears to be the first time. This opens the way for the use of characteristic mode theory in the design of antennas made of penetrable material whose polarization current densities constitute the main radiating mechanism of the antenna. Volume formulations are shown to be reliable but computationally burdensome. It is demonstrated that surface formulations are computationally more efficient, but obtrude some non-physical modes in addition to the physical ones. Fortunately, certain field orthogonality checklists can be used to provide a straightforward means of unambiguously selecting only the physical modes. The sub-structure characteristic mode concept is extended to problems involving both perfectly conducting and penetrable materials. It is also argued that sub-structure modes can be viewed as characteristic modes that implicitly use modified Green’s functions, but without such Green’s functions being needed explicitly. This makes the concept really practical, since the desired modified Green’s functions are not known explicitly in most cases.

Αντικείμενο της διατριβής είναι η επίλυση των προβλημάτων της σκέδασης επιπέδων ακουστικών κυμάτων χαμηλών συχνοτήτων από ένα διαπερατό σφαιρικό κέλυφος με έκκεντρο μαλακό, σκληρό ή διαπερατό πυρήνα και από μια μαλακή σφαίρα κάτω από ένα διαπερατό επίπεδο. Η λύση των προβλημάτων σκέδασης στην περιοχή χαμηλών συχνοτήτων επιδέχεται ανάπτυγμα Taylor σε δυνάμεις του κυματικού αριθμού k, όπου οι συντελεστές του αναπτύγματος (προσεγγίσεις χαμηλής συχνότητας) συνιστούν ακολουθία λύσεων στάσιμων προβλημάτων της θεωρίας δυναμικού. Ένα πρόβλημα σκέδασης μπορεί να δεχθεί προσέγγιση χαμηλών συχνοτήτων όταν το μήκος κύματος της κυματικής διαταραχής είναι πολύ μεγαλύτερο από την ακτίνα της ελάχιστης περιγεγραμμένης σφαίρας του σκεδαστή. Το δισφαιρικό σύστημα συντεταγμένων παρέχει κατάλληλο περιβάλλον για την επίλυση προβλημάτων πολλαπλής σκέδασης από δύο σφαίρες Αυτό ισχύει μόνο στη περιοχή των χαμηλών συχνοτήτων δεδομένου ότι η εξίσωση Laplace επιδέχεται διαμορφωμένο χωρισμό στις δισφαιρικές συντεταγμένες, ενώ δεν συμβαίνει το ίδιο στην εξίσωση Helmholtz. Προσαρμόζοντας το δισφαιρικό σύστημα συντεταγμένων στην δεδομένη γεωμετρία του κάθε προβλήματος απλουστεύεται η περιγραφή των χώρων που ορίζονται από το έκκεντρο σφαιρικό κέλυφος και οι σφαιρικές επιφάνειες του προβλήματός μας περιγράφονται από διαφορετικές τιμές της ίδιας συντεταγμένης μεταβλητής, ενώ ο απομακρυσμένος χώρος περιγράφεται από μια γειτονιά της αρχής των συντεταγμένων στο παραμετρικό χώρο των μεταβλητών η, θ. Επιλύοντας τα αντίστοιχα προβλήματα συνοριακών συνθηκών για μηδενική και πρώτης τάξεως προσεγγίσεις, καταλήγουμε σε αντίστοιχες αναγωγικές εξισώσεις ακολουθιών των συντελεστών ή αντίστοιχα συστήματα αναγωγικών εξισώσεων. Δεδομένου ότι οι ακολουθίες των συντελεστών συγκλείνουν ταχύτατα, περιοριζόμαστε στους πρώτους όρους συντελεστών και οι αναδρομικές εξισώσεις ή τα συστήματα αναγωγικών εξισώσεων ανάγονται σε εξισώσεις πινάκων ή γραμμικά συστήματα εξισώσεων με άγνωστους πίνακες στήλες και συντελεστές των αγνώστων τριδιαγώνιοι πίνακες. Με την πρωτότυπη αυτή μέθοδο προσδιορίζονται ακριβώς οι πρώτοι όροι χαμηλών συχνοτήτων των δύο προσεγγίσεων μηδενικής και οι πρώτης τάξεως, και στη συνέχεια οι προσεγγίσεις του πλάτους σκέδασης και των ενεργειακών διατομών σκέδασης. Μειώνοντας την απόσταση d των κέντρων συμπεραίνουμε ότι το πρόβλημα της σκέδασης ομόκεντρου σφαιρικού φλοιού δεν μπορεί να θεωρηθεί ειδική περίπτωση του προαναφερθέντος προβλήματος. / A plane wave is scattered by an acoustically soft, hard or penetrable sphere, covered by a penetrable non-concentric spherical lossless shell which disturbs the propagation of the incident plane wave field. There is exactly one bispherical coordinate system that fits the given two-sphere obstacle. If the wavelength of the incident field is much larger than the radius of the exterior sphere, Low Frequency Theory reduces the scattering problem to a sequence of potential problems which can be solved iteratively Applying the corresponding boundary value problem for each case, a set of two equations results as well as a recurrence equation with three unknown sequence of coefficients for zero-th order, and the first-order approximation is obtained, by solving two sets of two equations and a recurrence equation with three unknown sequence coefficients each for the soft core or the calculation of the zero–th order coefficients of the hard or penetrable core, leads to a solution of a linear system of two equations with two unknown columns and tri-diagonal square matrices are coefficients of the unknown columns, while the first-order approximation is obtained, by solving two linear systems of two equations with four unknown columns and eight tri-diagonal matrices as coefficients of the unknown columns. Applying the cut-off method for soft, hard and penetrable sphere, the low-frequency coefficients of the zero-th and first-order for the near field as well as the first and second-order coefficients are obtained for the normalized scattering amplitude and cross section. Decreasing the distance d of the centres we conclude that the problem of scattering concentric cell cannot be considered special case of mentioned before problem. A plane wave is scattered by an acoustical soft acoustic sphere embedded into an acoustically lossless half space, which disturbs the propagation of the incident wave field. In the first step, the problem of sound diffraction by only a penetrable plane is solved, were the amplitudes of reflective and diffractive acoustical waves are calculated. In the second step the diffractive as an incident wave is scattered by the embedded acoustical soft sphere. The low frequency zero-th and first order coefficients of the near field are calculated for the soft scatterer and finally the scattering amplitude and cross-section are determined.

<p>Soft matter systems exhibiting spatially modulated patterns on a mesoscale are characterized by many long-lived metastable phases for which relaxation to equilibrium is difficult and a satisfactory thermodynamic description is missing. Current dynamical theories suffer as well, because they mostly rely on an understanding of the underlying equilibrium behavior. This thesis relates the study of two canonical examples of modulated systems: microphase and cluster crystal formers. Microphases are the counterpart to gas-liquid phase separation in systems with competing short-range attractive and long-range repulsive interactions. Periodic lamellae, cylinders, clusters, etc., are thus observed in a wide variety of physical and chemical systems, such as multiblock copolymers, oil-water surfactant mixtures, charged colloidal suspensions, and magnetic materials. Cluster crystals in which each lattice site is occupied by multiple particles are formed in systems with steep soft-core repulsive interactions. Dendrimers have been proposed as a potential experimental realization. In order to access and understand the equilibrium properties of modulated systems, we here develop novel Monte Carlo simulation methods. A thermodynamic integration scheme allows us to calculate the free energy of specific modulated phases, while a [N]pT ensemble simulation approach, in which both particle number and lattice spacing fluctuate, allows us to explore their phase space more efficiently. With these two methods, we solve the equilibrium phase behavior of five schematic modulated-phase-forming spin and particle models, including the axial next-nearest-neighbor Ising (ANNNI) model, the Ising-Coulomb (IC) model, the square-well linear (SWL) model, the generalized exponential model of index 4 (GEM-4) and the penetrable sphere model (PSM). Interesting new physics ensues. In the ANNNI layered regime, simple phases are not found to play a particularly significant role in the devil's flowers and interfacial roughening plays at most a small role. With the help of generalized order parameters, the paramagnetic-modulated critical transition of the ANNNI model is also studied. We confirm the XY universality of the paramagnetic-modulated transition and its isotropic nature. With our development of novel free energy minimization schemes, the determination of a first phase diagram of a particle-based microphase former SWL is possible. We identify the low temperature GEM-4 phase diagram to be hybrid between the Gaussian core model (GCM) and the PSM. The system additionally exhibits S-shaped doubly reentrant phase sequences as well as critical isostructural transitions between face-centered cubic (FCC) cluster solids of different integer occupancy. The fluid-solid coexistence in the PSM phase diagram presents a crossover behavior around T~0.1, below which the system approaches the hard sphere limit. Studying this regime allows us to correct and reconcile prior DFT and cell theory work around this transition.</p> / Dissertation