Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Nonlinear Water Wave Computations Using a Multipole Accelerated, Desingularized Method
Nonlinear Water Wave Computations Using a Multipole Accelerated, Desingularized Method
Pages 64-74
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From page 64... ...
A BSTRACT Nonlinear inviscid water wave computations are performed using an Euler-Lagrange approach to solve a boundary integral formulation. The integral equations are solved numerically at each time step using a desingularized method with multipole acceleration.
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From page 65... ...
The efficiency of the multipole algorithm is evaluated by solving a fictitious boundary value problem for the [Cankine source strength using an iterative solver with and without multipole acceleration. Next, a new method for including fully nonlinear incident waves in the forward speed problem is introduced.
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From page 66... ...
The kinematic and dynamic free surface boundary conditions are then integrated in time. The mixed boundary value problem for the perturbation potential is determined by solving the Laplace equation in the fluid domain.
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From page 67... ...
are the spherical harmonics. Using the series solution to the Laplace equation, the far field potential due to a collection' of near field sources can be expressed in a multipole expansion.
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From page 68... ...
It is the careful arrangement of the clusters of panels which leads to the Nlog N effi cien cy The fast- multip ole algorithm redu ces the cost yet further, to order N by a complerocntar,, arrangement of the evaluation points so that accumulated multipole expansions may be transformed to local expansions centered in the clusters of evaluation points, and these expansions are evaluated instead.
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From page 69... ...
This solution procedure results in an O(N) method because the matrix A is no longer computed and stored and the multipole algorithm allows the linear system to be solved in O(N)
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From page 70... ...
Nonlinear incident waves at forward speed Because boundary conditions are required on all en closing surfaces for any boundary integral method, the problem of producing fully nonlinear incident waves on a ship moving at forward speed is not trivial. The dif ficulty arises because computational limitations allow only a portion of the free surface to be modeled.
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From page 71... ...
The downstream boundary is left open and the quip is infinitely deep. A small distance below tale free surface an axisymrnetric submarine Bull form is translating with steady forward speed UO The submarine hull form is described using the following formula from Jackson (1991~.
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From page 72... ...
The number of nodes required to adequately resolve the Kelvin wake would be very large. Since the forces resulting from steady forward speed are small relative to the Eroude-Krylov forces due to incident waves, we used the coarse grid expecting a small error due to inadequate resolution of the Kelvin wake.
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From page 73... ...
A new method has been presented for introducing nonlinear incident waves into problems with forward speed. This method can be used to produce waves inside both two- and three-dimensional computational windows.
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From page 74... ...
The remaining boundaries upstream, downstream and side all require appropriate boundary conditions. The examples we have published here have all been for ~ > 1/4 so the upstream condition is defined by the steady flow or by incident waves.
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