Second-Order Diffraction and Radiation of a Floating Body With Small Forward Speed

2010 ◽  
Author(s):  
Yan-Lin Shao ◽  
Odd M. Faltinsen
Keyword(s):  
Boundary Condition ◽  
Coordinate System ◽  
Traditional Method ◽  
Higher Order ◽  
Second Order ◽  
The Body ◽  
New Method ◽  
Forward Speed ◽  
Inertial Coordinate System ◽  
Body Boundary

The formulation of the second-order wave-current-body problem in the inertial coordinate system involves higher-order derivatives in the body boundary condition. A new method taking advantage of the body-fixed coordinate system in the near field is presented to avoid the calculation of higher-order derivatives in the body boundary condition. The new method has advantage over the traditional method when the body surface is with sharp corner or high curvature. The nonlinear wave diffraction and forced oscillation of floating bodies are studied up to second order in wave slope. A small forward speed is taken into account. The results of the new method are compared with that of the traditional method based on a formulation in the inertial coordinate system. When the traditional method applies, good agreement has been obtained.

Second-Order Diffraction and Radiation of a Floating Body With Small Forward Speed

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
Yan-Lin Shao ◽  
Odd M. Faltinsen
Keyword(s):  
Boundary Condition ◽  
Coordinate System ◽  
Traditional Method ◽  
Higher Order ◽  
Second Order ◽  
The Body ◽  
New Method ◽  
Forward Speed ◽  
Inertial Coordinate System ◽  
Body Boundary

The formulation of the second-order wave-current-body problem in the inertial coordinate system involves higher-order derivatives in the body boundary condition. A new method taking advantage of the body-fixed coordinate system in the near field is presented to avoid the calculation of higher-order derivatives in the body boundary condition. The new method has an advantage over the traditional method when the body surface has a sharp corner or high curvature. The nonlinear wave diffraction and forced oscillation of floating bodies are studied up to second order in wave slope. A small forward speed is taken into account. The results of the new method are compared with that of the traditional method based on a formulation in the inertial coordinate system. When the traditional method applies, good agreement has been obtained.

Large-Amplitude Transient Motion of Two-Dimensional Floating Bodies

1979 ◽  
Vol 23 (01) ◽  
pp. 20-31
Author(s):  
R. B. Chapman
Keyword(s):  
Boundary Condition ◽  
Wave Field ◽  
Large Amplitude ◽  
The Body ◽  
Body Motion ◽  
Source Distribution ◽  
Floating Body ◽  
Two Dimensional ◽  
Body Boundary ◽  
Free Mode

A numerical method is presented for solving the transient two-dimensional flow induced by the motion of a floating body. The free-surface equations are linearized, but an exact body boundary condition permits large-amplitude motion of the body. The flow is divided into two parts: the wave field and the impulsive flow required to satisfy the instantaneous body boundary condition. The wave field is represented by a finite sum of harmonics. A nonuniform spacing of the harmonic components gives an efficient representation over specified time and space intervals. The body is represented by a source distribution over the portion of its surface under the static waterline. Two modes of body motion are discussed—a captive mode and a free mode. In the former case, the body motion is specified, and in the latter, it is calculated from the initial conditions and the inertial properties of the body. Two examples are given—water entry of a wedge in the captive mode and motion of a perturbed floating body in the free mode.

Second Order Difference Frequency Wave-Current Loading Using Kelvin-Newman Approximation

2020 ◽  
Author(s):  
Farid P. Bakti ◽  
Moo-Hyun Kim
Keyword(s):  
Second Order ◽  
The Body ◽  
Difference Frequency ◽  
Computational Time ◽  
Forward Speed ◽  
Frequency Wave ◽  
Order Difference ◽  
First Order ◽  
Current Interaction ◽  
Current Loading

Abstract Kelvin & Newman introduced a linearization method to include the current (or forward speed) effect into the diffraction & radiation wave field for large-slender floating bodies. The K-N method assumes a steady far-field current while disregarding the steady potential field due to the presence of the body. The method is proven to be reliable when the Froude number is relatively small, the body shape is relatively slender (∂∂x≪∂∂y,∂∂z), and the sea condition is mild. This requirement is fulfilled for typical FPSOs and ship-shaped vessels in a typical current (or forward speed) condition. Several studies suggested that the presence of the current might change the first order hydrodynamic coefficients such as the first order diffraction force, added mass, and radiation damping. Currents also contributed to a change in the second-order slowly-varying drift force. However, the effect of current in the second-order difference-frequency force is yet to be investigated. By expanding the Kelvin-Newman approximation up to the second order, and solving the problem in the frequency domain, we can save computational time while expanding the accuracy of the scheme. The second order quadratic force is the main focus of this study, since it is the main contributor to the total second order difference frequency forces especially near the diagonal. By implementing the Kelvin-Newman wave current interaction approach up to the wave’s second order, we can assess the performance of the Kelvin-Newman wave current interaction formulation in various sea conditions.

Radiation and diffraction of second-order surface waves by floating bodies

1988 ◽  
Vol 196 ◽  
pp. 65-91 ◽  
Author(s):  
P. D. Sclavounos
Keyword(s):  
Free Surface ◽  
Linear Problem ◽  
Velocity Potential ◽  
Surface Condition ◽  
Second Order ◽  
The Body ◽  
Green Functions ◽  
Floating Bodies ◽  
Free Surface Condition ◽  
Body Boundary

The paper studies the radiation and diffraction by floating bodies of deep-water bichromatic and bidirectional surface waves subject to the second-order free-surface condition. A theory is developed for the evaluation of the second-order velocity potential and wave forces valid for bodies of arbitrary geometry, which does not involve the evaluation of integrals over the free surface or require an increased accuracy in the solution of the linear problem. Explicit sum- and difference-frequency ‘Green functions’ are derived for the radiation and diffraction problems, obtained from the solution of initial-value problems that ensure they satisfy the proper radiation condition at infinity. The second-order velocity potential is expressed as the sum of a particular and a homogeneous component. The former satisfies the non-homogeneous free-surface condition and is expressed explicitly in terms of the second-order Green functions. The latter is subject to the homogeneous free-surface condition and enforces the body boundary condition by the solution of a linear problem. An analysis is carried out of the singular behaviour of the second-order potential near the intersection of the body boundary with the free surface.

Convergence Properties of the Neumann-Kelvin Problem for a Submerged Body

1987 ◽  
Vol 31 (04) ◽  
pp. 227-234
Author(s):  
Lawrence J. Doctors ◽  
Robert F. Beck
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Body Condition ◽  
Numerical Experiments ◽  
Surface Condition ◽  
The Body ◽  
Convergence Properties ◽  
Submerged Body ◽  
Body Boundary ◽  
The Galerkin Method

The Neumann-Kelvin method for solving the flow past a body moving at a steady speed requires that the body boundary condition be satisfied exactly, while the free-surface condition is satisfied in a linearized sense. The solution is generally obtained by discretizing the body surface into panels, each of which has an unknown singularity strength. In the present work, two types of numerical experiments have been carried out. In the first approach, the body condition is satisfied at one point on each panel (the collocation method). In the second approach, the body condition is satisfied in an integrated sense on each panel (the Galerkin method). Improved convergence properties are demonstrated by the second approach.

On the Wave Resistance of a Submerged Spheroid

1973 ◽  
Vol 17 (01) ◽  
pp. 1-11
Author(s):  
César Farell
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Resistance ◽  
The Body ◽  
Spheroidal Harmonics ◽  
Survey Technique ◽  
Body Boundary ◽  
Significant Comparison ◽  
The Difference ◽  
Infinite Set

A solution to the problem of potential flow about a submerged prelate spheroid in axial horizontal motion beneath a free surface has been derived within the theory of infinitesimal waves, satisfying exactly the body boundary condition, and the wave resistance of the spheroid has been evaluated. The solution is in the form of a distribution of sources on the surface of the spheroid; the analysis yields an infinite set of equations for determining the coefficients of the expansion of the potential of the distribution in spheroidal harmonics. The difference between the present results for the wave resistance and those given by Havelock's approximation is found to be rather significant. Comparison with experimental wave resistance measurements obtained using the wake-survey technique shows agreement for Froude numbers between 0.35 and 0.40.

Rankine Source Method for Seakeeping Analysis in Shallow Water

2014 ◽  
Author(s):  
Alexander von Graefe
Keyword(s):  
Boundary Condition ◽  
Shallow Water ◽  
Reference Frame ◽  
The Body ◽  
Forward Speed ◽  
Flat Bottom ◽  
Source Method ◽  
Rankine Source ◽  
Perturbation Potential ◽  
Fixed Reference

Following Hachmann approach, Söding’s Rankine source method describes the steady perturbation potential in the body-fixed reference frame. This is beneficial for seakeeping predictions at forward speed, because error-prone m-terms do not occur in the boundary condition on the ship hull. Although similar terms arise in the boundary condition on the free surface, numerical inaccuracies in these terms have much less influence on the behaviour of the ship than in the traditional approach, where m-terms are evaluated on the hull. The periodic flow is linearised with respect to the amplitude of the incident wave, taking into account the steady ship wave. A problem arises for shallow water problems if the Hachmann approach is used. For the shallow water problem, the boundary condition ‘no flow through the bottom’ must be fulfilled. Usually, this is done using image sources. If the steady perturbation potential is described in the body-fixed reference frame, this is not possible directly. This paper extends the Hachmann approach to treat this issue. Following the usual procedure for steady flow calculation, the boundary condition at the flat bottom of the steady problem is realised using image sources. For seakeeping computations, the steady perturbation potential is split into two parts, modelled respectively by steady sources above and below the flat bottom. The first part is assumed to be constant in the body-fixed reference frame, whereas the second part is assumed to be constant in the reference frame of the mirror image of the ship. This extended Hachmann approach fulfils the boundary condition at the flat bottom analytically. Computed ship motions with and without forward speed in shallow water are validated by model test results from Flanders Hydraulics Research, Antwerp, and from TU Berlin.

Acoustic Reconstructing Method in Prolate Spheroidal Coordinate System

2011 ◽  
Vol 181-182 ◽  
pp. 914-918
Author(s):  
Cai Xia Zhu ◽  
Shi Yuan Wang ◽  
Rui Liang Yang
Keyword(s):  
Boundary Condition ◽  
Coordinate System ◽  
Acoustic Pressure ◽  
Form Solution ◽  
New Method ◽  
Application Domain ◽  
Prolate Spheroidal ◽  
Pressure Function ◽  
Infinite Domains ◽  
Pressure Boundary Condition

An acoustic reconstructing method in prolate spheroidal coordinate system is proposed in this paper. Firstly, the entire acoustic field in prolate spheroidal coordinate system is divided by some infinite domains and every domain has some nodes. Then pressure functions are determined by requiring the assumed-form solution to satisfy the pressure boundary condition at the measured points. Finally, once nodal pressures are specified, the acoustic pressure everywhere is completely determined due to the known pressure function. The efficiency and precision of reconstruction can be significantly enhanced and satisfactory reconstruction can be obtained with relatively few measurements using the new method. The new method proposed in this paper is suitable for the acoustic sources with a characteristic aspect ratio y:z close 1:1 and x at random, which extends the appropriate application domain of traditional acoustic reconstructing method largely.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Consistent co-rotational framework for Euler-Bernoulli and Timoshenko beam-column elements under distributed member loads

2021 ◽  
pp. 136943322098663
Author(s):  
Yi-Qun Tang ◽  
Wen-Feng Chen ◽  
Yao-Peng Liu ◽  
Siu-Lai Chan
Keyword(s):  
Coordinate System ◽  
Stiffness Matrix ◽  
Traditional Method ◽  
Local Coordinate System ◽  
Frame Structures ◽  
Global Coordinate System ◽  
Single Element ◽  
Geometrically Nonlinear ◽  
Geometrically Nonlinear Analysis ◽  
Geometric Stiffness

Conventional co-rotational formulations for geometrically nonlinear analysis are based on the assumption that the finite element is only subjected to nodal loads and as a result, they are not accurate for the elements under distributed member loads. The magnitude and direction of member loads are treated as constant in the global coordinate system, but they are essentially varying in the local coordinate system for the element undergoing a large rigid body rotation, leading to the change of nodal moments at element ends. Thus, there is a need to improve the co-rotational formulations to allow for the effect. This paper proposes a new consistent co-rotational formulation for both Euler-Bernoulli and Timoshenko two-dimensional beam-column elements subjected to distributed member loads. It is found that the equivalent nodal moments are affected by the element geometric change and consequently contribute to a part of geometric stiffness matrix. From this study, the results of both eigenvalue buckling and second-order direct analyses will be significantly improved. Several examples are used to verify the proposed formulation with comparison of the traditional method, which demonstrate the accuracy and reliability of the proposed method in buckling analysis of frame structures under distributed member loads using a single element per member.

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