### Abstract

Nodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick-like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time-dependent convection-diffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM.

Original language | English (US) |
---|---|

Pages (from-to) | 144-164 |

Number of pages | 21 |

Journal | International Journal for Numerical Methods in Fluids |

Volume | 61 |

Issue number | 2 |

DOIs | |

State | Published - Sep 20 2009 |

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### Keywords

- Algebraic transformation
- Coarse mesh methods
- Convection-diffusion
- Irregular-shaped elements
- Nodal integralmethod
- Poissonequation

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mechanics
- Applied Mathematics
- Mechanical Engineering
- Mechanics of Materials

### Cite this

*International Journal for Numerical Methods in Fluids*,

*61*(2), 144-164. DOI: 10.1002/fld.1949

**A nodal integral method for quadrilateral elements.** / Nezami, Erfan G.; Singh, Suneet; Sobh, Nahil; Uddin, Rizwan.

Research output: Contribution to journal › Article

*International Journal for Numerical Methods in Fluids*, vol 61, no. 2, pp. 144-164. DOI: 10.1002/fld.1949

}

TY - JOUR

T1 - A nodal integral method for quadrilateral elements

AU - Nezami,Erfan G.

AU - Singh,Suneet

AU - Sobh,Nahil

AU - Uddin,Rizwan

PY - 2009/9/20

Y1 - 2009/9/20

N2 - Nodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick-like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time-dependent convection-diffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM.

AB - Nodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick-like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time-dependent convection-diffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM.

KW - Algebraic transformation

KW - Coarse mesh methods

KW - Convection-diffusion

KW - Irregular-shaped elements

KW - Nodal integralmethod

KW - Poissonequation

UR - http://www.scopus.com/inward/record.url?scp=70349615610&partnerID=8YFLogxK

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U2 - 10.1002/fld.1949

DO - 10.1002/fld.1949

M3 - Article

VL - 61

SP - 144

EP - 164

JO - International Journal for Numerical Methods in Fluids

T2 - International Journal for Numerical Methods in Fluids

JF - International Journal for Numerical Methods in Fluids

SN - 0271-2091

IS - 2

ER -