Time Domain Simulation of Two Interacting Ships Advancing Parallel in Waves

2011 ◽  
Author(s):  
Xu Xiang ◽  
Odd M. Faltinsen
Keyword(s):  
Diffraction Problem ◽  
Radiation Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Wave ◽  
Rankine Source ◽  
Runge Kutta Method ◽  
Wave Effects ◽  
Initial Boundary Value ◽  
Numerical Beach

A 3D Rankine source method is developed to solve the initial-boundary value problem of two ships advancing in waves. Linear wave effects are considered. The two ships are assumed advancing parallel with identical forward speed, with or without stagger, and the Neumann-Kelvin flow is chosen as the steady basis flow. An artificial numerical beach is applied to satisfy the radiation condition. A fourth order Runge-Kutta method is used for the 12 degree of freedom dynamic solver of the two ships. The present solver is validated through studying linear radiation and diffraction problem of one or two ships by comparing with analytical or model test results. The coupled motion solver is applied to a single S-175 ship advancing in waves and two parallel advancing ships which were tested in the MARINTEK towing tank (Ronæss, 2002) and promising agreements are obtained.

Vibration of Elastic Continuum With Stochastically Varying Surface Roughness Carrying an Accelerating Oscillator

2001 ◽  
Author(s):  
Christian A. Schenk ◽  
Lawrence A. Bergman
Keyword(s):  
Surface Roughness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Distributed Parameter ◽  
Direct Integration ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Second Moment ◽  
Runge Kutta Method ◽  
Initial Boundary Value

Abstract The problem of calculating the second moment properties of the response of a general class of non-conservative linear distributed parameter systems with stochastically varying surface roughness excited by a moving concentrated load is investigated. In particular, the method as presented in [1] is extended to the case of an arbitrarily varying oscillator speed. The resulting initial boundary value problem is transformed into the modal state space, where the second moment characteristics of the response are determined by direct integration using a Runge-Kutta method.

On the Transversal Vibrations of a Conveyor Belt Using a String-Like Equation

2001 ◽  
Author(s):  
Gede Suweken ◽  
W. T. van Horssen
Keyword(s):  
Wave Speed ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conveyor Belt ◽  
Linear Wave ◽  
Long Time ◽  
Vertical Vibrations ◽  
Transversal Vibrations ◽  
Initial Boundary Value

Abstract In this paper an initial-boundary value problem for a linear wave (string) equation is considered. This problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed. In this paper the belt is assumed to move with varying speed. Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt. It also will be shown that for this problem, the truncation method is not valid on long time scales.

Refraction and Diffraction of Nonlinear Waves in Coastal Waters by the Level I Green-Naghdi Equations

2003 ◽  
Author(s):  
R. C. Ertekin ◽  
H. Sundararaghavan
Keyword(s):  
Coastal Waters ◽  
Water Waves ◽  
Wave Equations ◽  
Diffraction Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Wave Propagation ◽  
Shallow Water Wave Equations ◽  
Level I ◽  
Initial Boundary Value

The Green-Naghdi (GN) equations are shallow-water wave equations which can be solved to predict the nonlinear and dispersive effects of water waves propagating in coastal waters. In this study, we formulate the Level I GN equations for variable bathymetry in 3-D, and numerically model the refraction-diffraction problem as an initial-boundary-value problem. A finite-difference model, in conjunction with elliptic grid-generation, is developed to solve the GN equations. Nonlinear wave propagation over a varying bathymetry is presented for solitary and cnoidal waves. A number of benchmark cases have been tested and compared with the available theoretical predictions. The results are presented and discussed to reveal the accuracy of the present model.

scholarly journals A Wave Equation Associated with Mixed Nonhomogeneous Conditions: The Compactness and Connectivity of Weak Solution Set

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Nguyen Thanh Long ◽  
Le Thi Phuong Ngoc
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Initial Boundary Value

The purpose of this paper is to show that the set of weak solutions of the initial-boundary value problem for the linear wave equation is nonempty, connected, and compact.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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