A Calculation Method of Hydrodynamic Forces on a Shallow-Draft and Very Large Floating Structure
Masashi Kashiwagi
Research Institute for Applied Mechanics, Kyushu University
Keywords: Floating structure, Hydroelasticity, B-spline function, Galerkin scheme, Wave force
ABSTRACT
An effective numerical calculation scheme is developed for the pressure distribution method which can be used to compute hydroelastic responses of a very large and shallow-draft floating structure. Bi-cubic B-spline functions are used for representing the unknown pressures and a Galerkin scheme is employed for converting the integral equation into algebraic simultaneous equations. Careful consideration is paid to reduce the computation time and to make free from errors associated with singular integrals. Excellent performance of the scheme is confirmed by checking the Haskind relation and the energy-conservation principle extended to specified elastic mode shapes. Numerical convergence is also verified with increasing the number of panels.
1. INTRODUCTION
Barge-type very large floating structures are recently considered as a possible airport. The size of the airport under consideration will be of order of 5 km long and 1 km wide but the draft is very small compared to dimensions of the plan view. Therefore this type of structure will be very flexible and elastic deformations may be more important than the rigid-body motions. To estimate these deformations, the mode -expansion method (modal analysis) can be applied, in which first-order hydrodynamic forces corresponding to each of specified mode shapes must be accurately calculated in the region of relatively very short wavelengths. For example, if we consider a realistic wave of 50 in wavelength, the length ratio between wave and structure is 1/100.
A simple but direct method for achieving that purpose is the pressure distribution method, studied first by Yamashita1). Several authors, e.g. Ikoma et al.2)and Yago3), showed numerical results based on this method. However their accuracy in short wavelength region seems not enough, because the zero-th order approximation is adopted in the discretization of the integral equation for unknown pressures. Besides, with zero-th order panel method, the number of unknowns must be ο(105) for wavelengths to be considered, and thus the computation time will be far from practical.
We are required to develop a method which can compute very accurately with fewer unknowns and less computation time. In this paper, to reduce the number of unknowns, bi-cubic B-spline functions are employed for representing unknown pressures and only one quarter of the structure is analyzed, with hydrodynamic symmetry relations taken into account. Further, to increase the accuracy, a Galerkin scheme is applied when converting the integral equation into a linear system of simultaneous equations. This scheme, as a trade-off, is said toincrease the computation time. However, in the present case, we can use relative similarity relations in evaluating the influence coefficients in the matrix; which drastically reduces the computation time. To make free from errors associated with singular integrals originating from a Rankine source in the kernel function, analytical integrations are performed and incorporated in the scheme.
Numerical convergence of the results with increasing the number of panels is verified and explicitly shown for the added-mass and damping coefficients in some graphs. It seems that the phenomena due to irregular frequencies do not occur. Furthermore for each wavelength, numerical accuracy is checked through the energy-conservation principle and the Haskind relation extended to elastic modes of motion. Obtained results up to L/λ = 50 (L: structure length, λ: wavelength) are quite satisfactory, in terms of accuracy, computation time, and numbers of unknowns.
2. INTEGRAL EQUATION
The coordinate system is taken as in Fig. 1. The plan view of the structure is rectangle with length L and width B, and the draft is very small compared to L and B. The incident angle of incoming wave is denoted by β and the water depth is constant at ｚ = h. The fluid is assumed inviscid and its motion is irrotational, which justifies the use of the velocity potential.
Time-harmonic motions of small amplitude are considered, with the complex time dependence eiωｔ applied to all first-order oscillatory quantities. The boundary conditions on the body and free surface are linearized. Then
