Tube hydroforming process is a relatively new technology compared to conventional manufacturing via stamping and welding. However there is not much knowledge available for the product or process designers. The objective of this study will determine the flow stress and forming limit diagram of tubular materials to discuss the formability of tubes. Firstly, a mathematical model is proposed to examine the plastic deformation behavior of a thin-walled tube at different process parameters during the bulge hydroforming process without axial feeding. In the formulation of this mathematical model, an ellipsoidal surface and non-uniform thinning in the free bulged region and sticking friction between the tube and die are assumed. In the sticking friction mode, the elements after contact with the die do not move or slide. The effects of various forming parameters, such as the die entry radius, the bulge length, anisotropy, the initial thickness of the tube, etc., upon the forming pressures are discussed systematically. Secondly, an analytical model combined with hydraulic bulge tests is proposed to evaluate the properties of tubular materials considering anisotropy effect. Annealed AA6011 aluminum tubes and SUS409 stainless steel tubes are used for the bulge test. The tube thickness and radius at the pole and the internal forming pressure are measured simultaneously during the bulge test. The anisotropic values are obtained from tensile tests. From above experimental data, the effective stress - effective strain relations can be derived by this analytical model. The finite element method is used to conduct the simulations of hydraulic bulge forming with the flow stresses obtained by the above-mentioned model. The analytical forming pressures versus bulge heights are compared with the experimental results to validate the approach proposed in this study. Finally, this study also establishes the Forming Limit Diagram (FLD) of aluminum tubular material. An experimental system of tube hydroforming in which axial feed is applied to carry out the hydraulic bulge-forming test of the annealed aluminum alloy tubes. Furthermore, Hill¡¦s new yield criterion is also used to predict the Forming Limit Curves of sheets. The predicted forming limit diagrams are compared with the experimental data. The results of this study can provide useful knowledge for process design. In addition, the process parameters of flow stress and forming limit diagram obtained can improve the accuracy of the simulation results in industrial and academic fields.

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The determination of the figure of the Earth and its gravity field has long relied on methodologies that approximate the Earth by a sphere, but this level of accuracy is no longer adequate for many applications, due to the advent of new and advanced measurement techniques. New, practical and highly accurate methodologies for gravity field modelling that describe the Earth as an oblate ellipsoid of revolution are therefore required. The foundation for these methodologies is formed by solutions to ellipsoidal geodetic boundary-value problems. In this thesis, new solutions to the ellipsoidal Dirichlet, Neumann and second-order boundary-value problems, as well as the fixed- and free-geodetic boundary-value problems, are derived. These solutions do not rely on any spherical approximation, but are nevertheless completely based on a simple spherical harmonic expansion of the function that is to be determined. They rely on new relations among spherical harmonic base functions. In the new solutions, solid spherical harmonic coefficients of the desired function are expressed as a weighted summation over surface spherical harmonic coefficients of the data on the ellipsoidal boundary, or alternatively as a weighted summation over coefficients that are computed under the approximation that the boundary is a sphere. / Specific applications of the new solutions are the computation of geopotential coefficients from terrestrial gravimetric data and local or regional gravimetric geoid determination. Numerical closed-loop simulations have shown that the accuracy of geopotential coefficients obtained with the new methods is significantly higher than the accuracy of existing methods that use the spherical harmonic framework. The ellipsoidal corrections to a Stokesian geoid determination computed from the new solutions show strong agreement with existing solutions. In addition, the importance of the choice of the reference sphere radius in Stokes's formula and its effect on the magnitude and spectral sensitivity of the ellipsoidal corrections are pointed out.

In general relativity, the nature of mass is non-local. However, an appropriate def-inition of mass at a quasi-local level could give a more detailed characterization ofthe gravitational field around massive bodies. Several attempts have been made tofind such a definition. One of the candidates is the Hawking mass. This thesispresents a method for calculating the spin coefficients used in the expression for theHawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal2-surfaces in Minkowski spacetime. Furthermore, the Hawking mass is shown to havethe correct limits, both in Minkowski and Schwarzschild, along particular foliationsof leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are alsopresented.

We propose a clustering algorithm which incorporates a similarity-based fuzzy clustering and principal component analysis. The proposed algorithm is capable of discovering clusters with hyper-spherical, hyperellipsoidal, or oblique hyper-ellipsoidal shapes. Besides, the number of the clusters need not be specified in advance by the user. For a given dataset, the orientation, locations, and the number of clusters obtained can truthfully reflect the characteristics of the dataset. Experimental results, obtained by running on datasets generated synthetically, show that our method performs better than other methods.

<p>In general relativity, the nature of mass is non-local. However, an appropriate def-inition of mass at a quasi-local level could give a more detailed characterization ofthe gravitational field around massive bodies. Several attempts have been made tofind such a definition. One of the candidates is the Hawking mass. This thesispresents a method for calculating the spin coefficients used in the expression for theHawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal2-surfaces in Minkowski spacetime. Furthermore, the Hawking mass is shown to havethe correct limits, both in Minkowski and Schwarzschild, along particular foliationsof leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are alsopresented.</p>

Τα τελευταία χρόνια, γίνεται πολύς λόγος για τους καρκινικούς όγκους, καθώς η νόσος αυτή προσβάλλει ολοένα και περισσότερα άτομα κάθε χρόνο. Ιδιαίτερη βαρύτητα έχει δοθεί τόσο ερευνητικά όσο και ιατρικά στην αντιμετώπιση του καρκίνου μέσω θεραπευτικών τεχνικών (χημειοθεραπείες, χειρουργικές επεμβάσεις κλπ) καθώς και στην βελτίωση των συνθηκών διαβίωσης των καρκινοπαθών. Επίσης, αρκετή έμφαση στην έρευνα σχετικά με την ανάπτυξη του καρκίνου σε βιοχημικό επίπεδο και για την βαθύτερη κατανόηση της νόσου. Η έρευνα αφορά ερευνητές πολλών διαφορετικών ειδικοτήτων μεταξύ των οποίων και των μαθηματικών. Από το 1954 με την πρόταση των Armitage και Doll σχετικά με την μαθηματική μοντελοποίηση της γένεσης των καρκινικών όγκων, αρκετοί έχουν ασχοληθεί με την μαθηματική προτυποποίηση των διαφόρων φάσεων του καρκίνου, από την δημιουργία του μέχρι και την αντίσταση του σε φαρμακευτική αγωγή. Η εργασία αυτή πραγματεύεται την μαθηματική θεμελίωση και προτυποποίηση των καρκινικών όγκων όσον αφορά την γεωμετρική τους ανάπτυξη. Με βάση το θεμελιώδες μαθηματικό μοντέλο που προτάθηκε το 1976 από τον H. P. Greenspan, μελετάται η επίπτωση επιφανειακών διαταραχών στην ανάπτυξη σφαιρικών καθώς και ελλειψοειδών όγκων. Στην πρωτότυπη εργασία, η μελέτη περιορίστηκε στην ανάλυση των διαταραχών με μεταβλητή την πολική γωνία των σφαιρικών συντεταγμένων. Στην εργασία αυτή αρχικά μελετάται η γενίκευση του μοντέλου διαταραχών και στις δυο γωνίες του σφαιρικού συστήματος συντεταγμένων (πολική θ και αζιμουθιακή φ). Στην συνέχεια επεκτείνεται η μέθοδος σε τρία μοντέλα που γενικεύουν τις παραδοχές του αρχικού μοντέλου διατηρώντας την παραδοχή της σφαιρικής γεωμετρίας και μελετάται η ευστάθεια των αντίστοιχων επιφανειακών διαταραχών. Τέλος, μελετάται και η ευστάθεια του ίδιου προβλήματος στην ελλειψοειδή γεωμετρία επειδή η ανισοτροπία του ελλειψοειδούς σχήματος καθιστά πιο ρεαλιστική την προσέγγιση του πραγματικού σχήματος του καρκινικού όγκου. / The mathematical analysis of the tumour growth attracted a lot of interest in the last two decades. However, as of today no generally accepted model for tumour growth exists. This is due partially to the incomplete understanding of the related pathology as well as the extremely complicated procedure that guides the evolution of a tumour. Moreover, the growth of a tumour does depend on the available tissue surrounding the tumour and therefore it represents a physical case that is realistically modelled by ellipsoidal geometry. The remarkable aspect of the ellipsoidal shape is that it represents the sphere of the anisotropic space. It provides the appropriate geometrical model for any direction dependent physical quantity. In the present work we analyze the stability of a spherical tumour for four continuous models of an avascular tumour and the stability study of an ellipsoidal tumour. For all five models, conditions for the stability are stated and the results are implemented numerically. For the spherical cases, it is observed that the steady state radii that secure the stability of the tumour are different for each of the four models, and that results to differences in the stable and unstable modes. As for the ellipsoidal model, it is shown that, in contrast to the highly symmetric spherical case, where stability is possible to be achieved, there are no conditions that secure the stability of an ellipsoidal tumour. Hence, as in many physical cases, the observed instability is a consequence of the lack of symmetry.

Η βέλτιστη γεωμετρική προσομοίωση του εγκεφάλου επιτυγχάνεται με ένα ελλειψοειδές. Η ηλεκτροεγκεφαλογραφία (ΗΕΓ) αφορά τις μετρήσεις του ηλεκτρικού δυναμικού στην επιφάνεια του κρανίου που αναπτύσσεται από νευρωνικά ρεύματα στο εσωτερικό του εγκεφάλου. Στην παρούσα εργασία έχουν παραχθεί οι απαιτούμενες ελλειψοειδείς αρμονικές πέμπτου, έκτου και εβδόμου βαθμού. Στη συνέχεια, χρησιμοποιούμε αυτές τις νέες ελλειψοειδείς αρμονικές συναρτήσεις για να εκφράσουμε το πλήρες αναλυτικό δυναμικό της ΗΕΓ. H δομή της εργασίας είναι: Στο κεφάλαιο 1 περιγράφεται περιληπτικά η φυσιολογία ανάπτυξης δυναμικών στην περιοχή του εγκεφάλου και πως επιτυγχάνεται η μετάδοση αυτών των σημάτων μέσω των νευρώνων. Στο κεφάλαιο 2 περιγράφεται η ελλειψοειδής γεωμετρία καθώς και η μορφή του τελεστή του Laplace στο ελλειψοειδές σύστημα συντεταγμένων. Αναφερόμαστε στην επίλυση της εξίσωσης Lame, στον τρόπο κατασκευής των συναρτήσεων Lame και περιγράφουμε τις αντίστοιχες ελλειψοειδείς αρμονικές. Στο κεφάλαιο 3 καταγράφουμε τις ελλειψοειδείς αρμονικές βαθμού 3. Στα κεφάλαια 4,5,6 και 7 παράγουμε τις ελλειψοειδείς αρμονικές βαθμού 4,5,6 και 7, αντίστοιχα, όπου η εφαρμογή των οποίων στην ΗΕΓ αποτελεί και το θέμα της παρούσας εργασίας. Τέλος στο κεφάλαιο 8 χρησιμοποιούμε τις ελλειψοειδείς αρμονικές που κατασκευάσαμε για να επιτύχουμε την ακριβή συνιστώσα του ηλεκτρικού δυναμικού που ανήκει στον ελλειψοειδή αρμονικό υπόχωρο που γεννούν οι 64 πρώτες αρμονικές συναρτήσεις. / The best geometric simulation of the human brain is achieved by an ellipsoidal system. The Electroencephalography (EEG) concerns the measurements of the electric potential on the surface of the head, which is generated from neuronal current inside the head. In this thesis we have produced the necessary ellipsoidal harmonics of the fifth, sixth and seventh degree. Then, we use these new ellipsoidal harmonic functions to express the full analytical potential of EEG, up to the seventh degree. The present thesis is structured in the following way: In chapter 1 we describe, in short, the physiology which develops the potentials in and outside the brain and how they are transmitted through the neuronal current. In chapter 2 we describe the ellipsoidal geometry as well as the form of the Laplace operator in ellipsoidal coordinates. We consider the solutions of Lame equation, the way Lame constructed them and we describe the corresponding ellipsoidal harmonics. In chapter 3 we report the ellipsoidal harmonics of degree zero through three. In chapters 4, 5, 6 and 7 we develop the ellipsoidal harmonics of degree 4, 5, 6 and 7 respectively. The applications of these functions form the main part of the present master thesis. Finally, in chapter 8 we use all the known ellipsoidal harmonics to express the relative component of the electrical potential. This component belongs to the subspace generated by the first 64 ellipsoidal harmonics.

To circumvent some inherent problems in the conventional ECG, this research reinvestigates an ‘unassisted’ approach which enables ECG measurement without the placement of leads on the body. Employed in this research is a widely accepted assumption that the electrical activity of the heart may be represented, largely, by a 3-D time-varying current dipole (3D-CD). From the PhysioBank database, mECG and fECG data were obtained, and Singular Value Decomposition (SVD) was performed to estimate the time-varying Vector ECG dipole. To determine the sensing matrix responsible for transforming the activity of the 3D-CD into the potential distribution on the surface of the medium, the ECG vector dipole signals are used to excite a 3D-CD in water medium of a specific shape-containing-ellipsoid model(s) in COMSOL tool. The sensing matrix thereby estimated is then utilized to reconstruct the 3D-CD signals from the signals measured by the probes on the surface of the medium. Fairly low NRMSEs (Normalized Root-Mean-Squared Errors) are attained. The approach is also successfully extended to the case of two ellipsoids, one inside the other, representing a pregnant female subject. Low NRMSEs (Normalized Root-Mean-Squared Errors) are again observed.

The aim of this diploma thesis is to design a suitable reflector concentrating a light beam into a circular hole in a gobo. The light beam behind the screen should have good efficiency and at the same time good homogeneity. For the reflector design and analysis a software for designing optical systems called LightTools was used. The reflector was designed by six different methods. Several smooth and several faceted reflectors were designed. The selected reflector was analyzed in combination with a varifocal lens. For selected reflector was designed mechanical store with respect to the anticipated operating conditions. This thesis serves as the basis for the producing of the reflector.