­^¤åºK­n A time-independent finite-difference numerical scheme is developed to study the dynamic response of a submerged and a floating breakwater under the wave loading of a fully numerical force. The coupled surge, heave and pitch motion of a floating breakwater and the wave-structure interaction are included in the model. The numerical results are validated uses several bench mark studies and results available elsemlse. The wave reducing effect of a submerged and a floating breakwaters were analysis and discusse.

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In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form \begin{align} F\left[D^2u-A(\,\cdot\,,u,Du)\right] &= B(\,\cdot\,,u,Du),\quad\text{in}\ \Omega^-,\notag\\ T_u(\Omega^-) &= \Omega^+, \end{align} where $A$, $B$, $T_u$ are all prescribed; and $\Omega^-$ along with $\Omega^+$ are bounded in $\mathbb{R}^n$, smooth and satisfying notions of c-convexity and c^*-convexity relative to one another (see [MTW05] for definitions). In particular, the case where $F$ is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F, introducing some complications that are not present in the Optimal Transportation case.¶ In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres.

A 2D fully nonlinear Numerical Wave Tank (NWT) is developed based on the potential theory, mixed Eulerian-Lagrangian (MEL) time marching scheme, and boundary element method (BEM). Nonlinear Wave deformation and wave forces on stationary and freely floating bodies are calculated using the NWT. For verification, the computed mean, 1st, 2nd, and 3rd order wave forces on a single submerged cylinder are compared with those of Chaplin's experiment, Ogilvie's 2nd-order theory, and other nonlinear computation called high-order spectral method. Similar calculations for dual submerged cylinders are also conducted. The developed fully nonlinear NWT is also applied to the calculations of the nonlinear pressure and force of surface piercing barge type structures and these obtained results agree with experimental and theoretical results. Nonlinear waves generated by prescribed body motions, such as wedge type wave maker or land sliding in the coastal slope area, can also be simulated by the developed NWT. The generated waves are in agreement with published experimental and numerical results. Added mass and damping coefficients can also be calculated from the simulation in time domain. For the simulation of freely floating barge-type structure, only fully nonlinear time-stepping scheme can accurately produce nonlinear body motions with large floating body simulations. The acceleration potential method, which was developed by Tanizawa (1996), is known to be the most accurate, consistent and stable. Using acceleration potential method, in the present study, the series of motions and drift forces were calculated over a wide range of incident wave frequencies including resonance region. To guarantitatively compare the nonlinear contribution of free-surface and body-boundary conditions, the body-nonlinear-only case with linearized free-surface condition is separately simulated. All the floating body motions and forces are in agreement with experimental results. Finally, the NWT is extended to fully nonlinear wave-body-current interactions of freely floating bodies, which has not been published in the open literature until now.

This study is to develop a 2D fully nonlinear numerical wave tank used to investigate the wave-induced dynamic properties of a dual pontoon floating structure (DPFS) with a liquid container on the top. The nonlinear numerical wave tank, developed based on the velocity potential function and the boundary element method (BEM), is to simulate dynamic properties including sway, heave, roll, and tension response. In addition, a physical model of the dual floating pontoon is tested in a hydrodynamic wave tank to validate the numerical model for simulation of wave and structure interaction. In the numerical model, a boundary integral equation method (BIEM) with linear element scheme is applied to establish a 2D fully nonlinear numerical wave tank (NWT). The nonlinear free surface condition is treated by combining the Mixed Eulerian and Lagrangian method (MEL), the fourth-order Runge-Kutta method (RK4) and a cubic spline scheme. The second-order Stokes wave theory is used to generate the velocity flux on the input boundary. Numerical damping zones are deployed at both ends of the NWT to dissipate or absorb the transmitted and reflected waves. Acceleration potential method and modal decomposition method are adopted to solve the unsteady potential functions £X1,t and £X2,t, while the system of motion equation is established according to Newton's 2nd law. Finally, the RK4 is applied to predict the motion of the platform, and the variation of free surface. As for the hydrodynamic laboratory model test, an image process scheme is applied to trace the floating structure motion and the variation of water surface inside the sloshing tank, while the mooring tension is measured by a load cell and stored in a data logger. The comparisons of numerical simulations and experimental data indicate that the numerical predictions are larger than measurements especially near the resonance frequency. This discrepancy is probably due to the fluid viscous effect. To overcome this problem and maintain the calculation efficiency, an uncoupled damping coefficient obtained through a damping ratio (£a=C/Ccr=0.02) is incorporated into the vibration system. Results reveal that responses of body motion near the resonant frequencies of each mode have significantly reduced and close to the measurements. Therefore, the suitable value of the damping ratio for the floating platform is £a=0.02. Then the numerical model with a damping ratio is applied to investigate the dynamic properties of the floating platform for different arrangements, including different mooring angle, spring constant, spacing, and the liquid container. Results demonstrate that the resonant frequency of each mode, responses of body motion and mooring tensions change along with the settings. As a whole, the platform with smaller mooring angle, longer spacing between the pontoons, higher water depth and wider width of the liquid container has relatively stable body motions and less mooring tension. Finally, the comparisons of the effects of random and regular waves on the floating structure illustrate that the variation of water surface in the liquid container is much severe in random waves than in regular waves such that the interaction between liquid and floating structure is more chaotic and thus reduces the amplitude of each response mode. As a result, the mooring tensions for random waves become much gentler than the regular waves. Key words: Boundary integral equation method; fully nonlinear numerical wave tank; dual pontoon floating structure

A 2D fully nonlinear Numerical Wave Tank (NWT) is developed based on the potential theory, mixed Eulerian-Lagrangian (MEL) time marching scheme, and boundary element method (BEM). Nonlinear Wave deformation and wave forces on stationary and freely floating bodies are calculated using the NWT. For verification, the computed mean, 1st, 2nd, and 3rd order wave forces on a single submerged cylinder are compared with those of Chaplin's experiment, Ogilvie's 2nd-order theory, and other nonlinear computation called high-order spectral method. Similar calculations for dual submerged cylinders are also conducted. The developed fully nonlinear NWT is also applied to the calculations of the nonlinear pressure and force of surface piercing barge type structures and these obtained results agree with experimental and theoretical results. Nonlinear waves generated by prescribed body motions, such as wedge type wave maker or land sliding in the coastal slope area, can also be simulated by the developed NWT. The generated waves are in agreement with published experimental and numerical results. Added mass and damping coefficients can also be calculated from the simulation in time domain. For the simulation of freely floating barge-type structure, only fully nonlinear time-stepping scheme can accurately produce nonlinear body motions with large floating body simulations. The acceleration potential method, which was developed by Tanizawa (1996), is known to be the most accurate, consistent and stable. Using acceleration potential method, in the present study, the series of motions and drift forces were calculated over a wide range of incident wave frequencies including resonance region. To guarantitatively compare the nonlinear contribution of free-surface and body-boundary conditions, the body-nonlinear-only case with linearized free-surface condition is separately simulated. All the floating body motions and forces are in agreement with experimental results. Finally, the NWT is extended to fully nonlinear wave-body-current interactions of freely floating bodies, which has not been published in the open literature until now.

This paper is to investigate the wave-induced dynamic properties of the floating platform for cage aquaculture. Considering the calculation efficiency and its applicability, this problem is simplified by: (1) assuming the flow field is inviscid, incompressible and irrotational; (2) the form drag and inertia drag on the fish net is calculated by the modified Morison equation (or Morison type equation of relative motion), including the material and geometric properties; (3) the moorings is treated as a symmetric linear spring system and the influence of hydrodynamic forces on the mooring lines is neglected; and (4) the net-volume is assumed as un-deformable to avoid the inversely prolonging computing time because the mass of fish net with is too light comparing with the mass of floating platform and cause the marching time step tremendously small to reach the steady-state condition which may lead to larger numerical errors (e.g. truncation errors) in computation. The BIEM with linear element scheme is applied to establish a 2D fully nonlinear numerical wave tank (NWT). The nonlinear free surface condition is treated by combining the Mixed Eulerian and Lagrangian method (MEL), the fourth-order Runge-Kutta method (RK4) and the cubic spline scheme. The second-order Stokes wave theory is adopted to give the velocity on the input boundary. Numerical damping zones are deployed at both ends of the NWT to dissipate or absorb the transmitted and reflected wave energy. The velocity and acceleration fields should be solved simultaneously in order to obtain the wave-induced dynamic property of the floating platform. Thus, both the acceleration potential method and modal decomposition method are adopted to solve the wave forces on the floating body, while the wave forces on the fish net are calculated by the modified Morison equation. According to Newton¡¦s second law, the total forces on the gravity center of the floating platform form the equation of motion. Finally, the RK4 is applied to predict the displacement and velocity of the platform. Firstly, the NWT is validated by comparing the wave elevation, internal velocity and acceleration with those from the second-order Stokes wave theory. Moreover, the numerical damping zone is suitable for long time simulation with a wide range of wave depth. The simulated results on wave-body interactions of fixed or freely floating body also indicate good agreement with those of other published results. Secondly, in the case of the interaction of waves and the floating platform, the simulated results show well agreement with experimental data, except at the vicinity of resonant frequency of roll and heave motions. This discrepancy is due to the fluid viscous effect. To overcome this problem and maintain the calculation efficiency, an uncoupled damping coefficient obtained by a damping ratio (£i=0.1 ) is incorporated into the vibration system. Results reveal that responses of body motion near the resonant frequencies of each mode have significant reduction and close to the experimental data. Moreover, the results are also consistent well with experiments in different wave height, mooring angle, water depth either with or without fish net. Therefore, the suitable value of the damping ratio for the floating platform is £i=0.1. Finally, the present model is applied to investigate the dynamic properties of the floating platform under different draft, width, spacing, spring constant, mooring angle and depth of fish net. Results reveal that the resonant frequency and response of body motion, mooring force, reflection and transmission coefficients and wave energy will be changed. According to the resonant response, the platform with shallower draft, larger width, longer spacing between two pontoons, smaller spring constants, or deeper depth of fish net has more stable body motions and smaller mooring forces. Irregular wave cases are presented to illustrate the relationship with the regular wave cases. Results indicate that the dynamic responses of body motion and the reflection coefficient in irregular waves have similar trend with regular waves. However, in the irregular wave cases, the resonant frequency is moved to the higher frequency. Similarly, resonant response function is smaller but wider, which is due to the energy distribution in the wave spectrum.

It is the main goal of this thesis to study the regularity of solutions for a nonlinear elliptic system coming from population segregation, and the free boundary problem that is obtained in the limit as the competition parameter goes to infinity [mathematical symbol]. The main results are existence and Hölder regularity of solutions of the elliptic system, characterization of the limit as a free boundary problem, and Lipschitz regularity at the boundary for the limiting problem. / text

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / In this paper we provide a universal solution for continuity module in the direction of the viscosity of fully nonlinear elliptic equations considering properties of the function f integrable in different situations. Established inner estimate for the solutions of these equations based on some conditions the norm of the function f. To obtain regularity in solutions of these inhomogeneous equations and coefficients of variables we use a method of compactness, which consists essentially of approximating solutions of inhomogeneous equations for a solution of a homogeneous equation in order to "inherit" the regularity that those equations possess. / Neste trabalho fornecemos mÃdulo de continuidade universal para soluÃÃes, no sentido da viscosidade,de equaÃÃes elÃpticas totalmente nÃo lineares, considerando propriedades de integrabilidade da funÃÃo f em diferentes situaÃÃes. Estabelecemos estimativa interior para as soluÃÃes dessas equaÃÃes baseadas em algumas condiÃÃes da norma da funÃÃo f. Para se obter regularidade nas soluÃÃes dessas equacÃes nÃo homogÃneas e de coeficientes variÃveis usamos um mÃtodo de compacidade, o qual consiste, essencialmente, em aproximar soluÃÃes de equaÃÃes nÃo homogÃneas por uma soluÃÃo de uma equaÃÃo homogÃnea com o objetivo de âherdarâ a regularidade que essas equaÃÃes possuem.

On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text