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Strathprints serves world leading Open Access research by the University of Strathclyde, including research by the Strathclyde Institute of Pharmacy and Biomedical Sciences (SIPBS), where research centres such as the Industrial Biotechnology Innovation Centre (IBioIC), the Cancer Research UK Formulation Unit, SeaBioTech and the Centre for Biophotonics are based.

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Numerical instability in linearized planing problems

Wang, Xuelian and Day, Alexander H. (2007) Numerical instability in linearized planing problems. International Journal for Numerical Methods in Engineering, 70 (7). pp. 840-875. ISSN 0029-5981

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Abstract

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.