# Numerical instability in linearized planing problems

Xuelian Wang, Alexander H. Day

Research output: Contribution to journalArticle

10 Citations (Scopus)

### 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.
Language English 840-875 35 International Journal for Numerical Methods in Engineering 70 7 10.1002/nme.1913 Published - 2007

### Fingerprint

Numerical Instability
Linear equations
Ships
Ship
Pressure distribution
Pressure Distribution
Linear equation
Hydrodynamics
Singular value decomposition
LU decomposition
Boundary element method
Gaussian elimination
Boundary value problems
Decomposition Techniques
Pyramid
System of Linear Equations
Collocation
Algebraic Equation
Boundary Elements
Overlapping

### Keywords

• boundary element method
• linearized planing theory
• numerical instability
• singular value decomposition
• marine engineering

### Cite this

@article{4f692b1f50704a4cb85b8093589ace53,
title = "Numerical instability in linearized planing problems",
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.",
keywords = "boundary element method, linearized planing theory, numerical instability, singular value decomposition, marine engineering",
author = "Xuelian Wang and Day, {Alexander H.}",
year = "2007",
doi = "10.1002/nme.1913",
language = "English",
volume = "70",
pages = "840--875",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
number = "7",

}

In: International Journal for Numerical Methods in Engineering, Vol. 70, No. 7, 2007, p. 840-875.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Numerical instability in linearized planing problems

AU - Wang, Xuelian

AU - Day, Alexander H.

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

KW - boundary element method

KW - linearized planing theory

KW - numerical instability

KW - singular value decomposition

KW - marine engineering

UR - http://dx.doi.org/10.1002/nme.1913

U2 - 10.1002/nme.1913

DO - 10.1002/nme.1913

M3 - Article

VL - 70

SP - 840

EP - 875

JO - International Journal for Numerical Methods in Engineering

T2 - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 7

ER -