Neste trabalho, estudamos propriedades de continuação única para as soluções da equação tipo Schrödinger com um ponto interação centrado em x=0, \\partial_tu=i(\\Delta_Z+V)u, onde V=V(x,t) é uma função de valor real e -\\Delta_Z é o operador escrito formalmente como \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] sendo \\delta_0 a delta de Dirac centrada em zero e Z qualquer número real. Logo, usamos estes resultados para ver o possível fenômeno de concentração das soluções, que explodem, da equação de tipo Schrödinger não linear com um ponto de interação em x=0, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] com ho>5. Também, mostramos que para certas condições sobre o potencial dependente do tempo V, a equação linear em cima tem soluções não triviais. / In this work, we study unique continuation properties for solutions of the Schrödinger equations with an point interaction centered at $x=0$, \\begin\\label \\partial_tu=i(\\Delta_Z+V)u, \\end where $V=V(x,t)$ is real value function and $-\\Delta_Z$ is the operator formally written \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] and $\\delta_0$ is Dirac\'s delta centered at zero and $Z$ is a real number. Next, we use these results in order to study the possible profile of the concentration of blow up solutions for the non linear Schrödinger equation with a point interaction at $x=0$, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] with $ho>5$. Besides, we show that the equation above has non trivial solutions for some conditions on the time dependent potencial $V$.

Neste trabalho, estudamos propriedades de continuação única para as soluções da equação tipo Schrödinger com um ponto interação centrado em x=0, \\partial_tu=i(\\Delta_Z+V)u, onde V=V(x,t) é uma função de valor real e -\\Delta_Z é o operador escrito formalmente como \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] sendo \\delta_0 a delta de Dirac centrada em zero e Z qualquer número real. Logo, usamos estes resultados para ver o possível fenômeno de concentração das soluções, que explodem, da equação de tipo Schrödinger não linear com um ponto de interação em x=0, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] com ho>5. Também, mostramos que para certas condições sobre o potencial dependente do tempo V, a equação linear em cima tem soluções não triviais. / In this work, we study unique continuation properties for solutions of the Schrödinger equations with an point interaction centered at $x=0$, \\begin\\label \\partial_tu=i(\\Delta_Z+V)u, \\end where $V=V(x,t)$ is real value function and $-\\Delta_Z$ is the operator formally written \\[-\\Delta_Z=-\\frac\\frac{d^2}{dx^2}+Z\\delta_0,\\] and $\\delta_0$ is Dirac\'s delta centered at zero and $Z$ is a real number. Next, we use these results in order to study the possible profile of the concentration of blow up solutions for the non linear Schrödinger equation with a point interaction at $x=0$, \\[\\partial_tu=i(\\Delta_Zu+|u|^u),\\] with $ho>5$. Besides, we show that the equation above has non trivial solutions for some conditions on the time dependent potencial $V$.

A high-order hybrid discontinuous Galerkin finite element method (DG-FEM) is developed for multi-layered curved panels having large deformation and finite strain. The kinematics of the multi-layered shells is presented at first. The Jacobian matrix and its determinant are also calculated. The weak form of the DG-FEM is next presented. In this case, the discontinuous basis functions can be employed for the displacement basis functions. The implementation details of the nonlinear FEM are next presented. Then, the Consistent Orthogonal Basis Function Space is developed. Given the boundary conditions and structure configurations, there will be a unique basis function space, such that the mass matrix is an accurate diagonal matrix. Moreover, the Consistent Orthogonal Basis Functions are very similar to mode shape functions. Based on the DG-FEM, three dedicated finite elements are developed for the multi-layered pipes, curved stiffeners and multi-layered stiffened hydrofoils. The kinematics of these three structures are presented. The smooth configuration is also obtained, which is very important for the buckling analysis with large deformation and finite strain. Finally, five problems are solved, including sandwich plates, 2-D multi-layered pipes, 3-D multi-layered pipes, stiffened plates and stiffened multi-layered hydrofoils. Material and geometric nonlinearities are both considered. The results are verified by other papers' results or ANSYS. / Master of Science / A novel computational method is developed for the composite structures withmultiple layers and stiffeners, which possess high ratio of strength-to-weight andhave wide applications in the aerospace engineering. The present method has thepotential to use fewer calculations to obtain high accuracy. Five typical andimportant problems are solved by this method and the results are also verifiedbyother papers or commercial software. For the first problem, the Sandwichplateproblem, the water pressure is applied on the top surface and the deformationaswell as stress field are both analyzed. The second problem is a two-dimensional multi-layered pipe’s collapse. The critical collapse failure point is found as a functionof geometrical imperfection. The third problem is the three-dimensional multilayered pipe’s unstable deformation analysis. The critical point of the unstabledeformation is found and a device is also analyzed to increase the strength. For thelast two problems, they are the stiffened plates and shells. In this case, weusestiffeners to increase the strength of the structure and the deformationof thestiffened plates/shells is analyzed. For the stiffened plate problem, we analyzearectangular plate reinforced by a parabolic stiffener. For the stiffened shell problem, we analyze the airfoil/hydrofoil structure stiffened by ribs. All these problems areimportant for aerospace vehicles.

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.