Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Computations of Wave Loads Using a B-Spline Panel Method
Computations of Wave Loads Using a B-Spline Panel Method
Pages 75-92
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From page 75... ...
A more fundamental restriction is that the potential derived from Green's theorem cannot be differentiated analytically to derive the fluid velocity on or near the body surface; this difficulty can be circumvented in a limited manner by using a distribution of sources only, but that method also fails if gradients of the velocity field are required. Higher-order panel methods have been devel7s
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From page 76... ...
It should be noted that we refer to 'order' in two completely different contexts here, one specifying the numerical approximation of the geometry and velocity potential on the body surface, and the other referring to the order of the perturbation expansion in terms of the amplitude of the waves and body motions. Low-order and higherorder panel methods are distinguished by the type of numerical approximations used, whereas firstand higher-order wave loads are defined with respect to the corresponding powers of the wave amplitude.
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From page 77... ...
(6) Each of the diffraction potentials also satisfies the condition of zero normal velocity on the body surface, and vanishes at large depths.
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From page 78... ...
The details of this procedure, and the utilization of B-spline basis functions are discussed in Section 3.3. The integrated wave loads or exciting forces are evaluated by integrating the pressure due to the diffraction potentials over the wetted surface of the body.
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From page 79... ...
On the pontoons a total of 32 patches are used, and outlined separately in the figure. Three larger patches are used to represent the column, including one on the bottom circular disk, one on the side to represent the complete circular cylinder above the pontoons, and one on the lower outside surface of the cylinder between the pontoons.
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From page 80... ...
With this algorithm, the field points of the Green function in the inner integration are specified by the Gauss nodes in parametric space. Special attention is required for the inner integrals, to account for the Rankine singularity of the Green function.
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From page 81... ...
As noted in §2, an appropriate procedure is to use numerical integration inside a circle of finite radius, and a semi-analytic analysis in the far field outside the same circle. Since the inner integration is the most difficult computational task in the low-order panel method, we concentrate on it here and truncate the free surface at a specified partition radius r, and neglect the far field contribution in the forcing function.
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From page 82... ...
panel method. To demonstrate this we first show results front WAMIT where the diffraction potential and its radial derivative are evaluated on the free surface, and compared with the analytical solution for a vertical circular cylinder.
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From page 83... ...
More accurate potential representations are obtained using knot vectors which subdivide the intervals between consecutive geometry knots. We define panels as the physical surface corresponding to the parametric Quantity wave amplitude cylinder radius half-distance between bodies acceleration due to gravity cylinder draft wavelength wavenumber fluid density radian frequency Table 1: Definition of commonly used symbols.
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From page 84... ...
Figure 8 compares the first-order surge exciting force on the TLP as computed by the present method and by the low-order panel method WAMIT. The graphical agreement with WAMIT is clear.
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From page 85... ...
To address the issue of computational efficiency, Table 2 compares the number of unknowns required and execution time, for equivalent accuracy, using HIPAN and using the conventional low-order panel method. Two applications are included in this comparison, the heave exciting force on a single truncated circular cylinder and the vertical mean drift force on a submerged sphere.
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From page 86... ...
. The results in the upper table are for the surge exciting force on a truncated circular cylinder, and in the lower table for the vertical second-order drift force on a submerged sphere in finite depth.
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From page 87... ...
We are using HIPAN to develop a more general solution of the third-order problem which is applicable to practical body shapes including the TLP. This computational approach can be developed following the boundary conditions and assumptions of either FNV or M&M, so that it is possible to develop independent results for comparison with their more analytical solutions.
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From page 88... ...
The accuracy and efficiency of 13-splines means that the description of the geometry can be practically exact, and the representations of the velocity potential and its derivatives are continuous. Thus wave radriation and diffraction problems can be analyzed more efficiently than with the low-order panel method, and it is possible to develop second- and third-order solutions with a more robust evaluation of the inhomogeneous forcing function on the free surface.
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From page 89... ...
An important practical consideration is the difficulty of using HIPAN. Unlike low-order panel methods where the geometry is described by a set of vertices, or points in Cartesian space situated on the body surface, HIPAN requires a set of Bspline control points which are relatively small in number, but more abstract to interpret.
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From page 90... ...
B & Maniar, H., "Calculating the double body m-terms with a higher order B-spline based panel method", Eleventh International Workshop on Water Waves arid Floating Bodies.
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From page 91... ...
Our own experience with the low-order panel method also suggested that putting a "lid" on the interior free surface is effective from the computational point of view. We have refined this technique and used it in the low-order panel code WAMIT to remove the irregular frequency effects from the first- and second-order nonlinear solution.
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From page 92... ...
2. The lines midway between cylinclers look like appropriate locations to apply periodic boundary conditions if the wavenumber in the in-line cylinder direction is properly chosen.
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