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Numerical instability in linearized planing problems
Author(s) -
Wang Xuelian,
Day Alexander H.
Publication year - 2006
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1913
Subject(s) - mathematics , discretization , algebraic equation , finite element method , mathematical analysis , boundary value problem , singular value decomposition , collocation (remote sensing) , system of linear equations , nonlinear system , computer science , physics , structural engineering , engineering , algorithm , quantum mechanics , machine learning
The hydrodynamics of planing ships are studied using a finite pressure element method. In this method, a boundary value problem (BVP) is formulated based on linear planing theory; the planing ship is represented by the pressure distribution acting on the wetted bottom of the ship, and the magnitude of this pressure distribution is evaluated using a boundary element method. The pressure is discretized using overlapping pressure pyramids, known as tent functions, so that the resulting distribution is piece‐wise continuous in both longitudinal and transverse directions. A set of linear algebraic equations for the determination of the pressure is then established using a collocation technique. It is found that the matrix of the linear equations is ill conditioned; this leads to oscillatory behaviour of the predicted pressure distribution if the direct solution method of LU decomposition or Gaussian elimination is used to solve the system of linear equations. In the current study, this numerical instability is analysed in detail. It is found that the problem can be addressed by adopting singular value decomposition (SVD) technique for the solution of the linear equations. Using this method, the hydrodynamic results for flat‐bottomed and prismatic planing ships are calculated and a good agreement is demonstrated with Savitsky's empirical relations. Copyright © 2006 John Wiley & Sons, Ltd.