### Abstract

One of the main advantages of meshless methods is that it eliminates the mesh generation, but it is still necessary to place nodes with controlled spacing variation on the boundary and within the domain. However, due to lack of connectivity between nodes it is more difficult to interpolate the field variables and impose boundary conditions. In this paper, a mesh free method is presented for analysis using a structured grid that does not conform to the geometry of the domain. The geometry of the domain is independent of the structured grid and is represented using implicit equations. The implicit equations of the boundaries can be used to construct solution structures that satisfy boundary conditions exactly even though the nodes of the grid are not on the boundaries of the domain. The solution structures are constructed using the implicit equations of the boundary together with a piece-wise interpolation over the structured grid. The implicit equations are also used to construct step function of solid such that its value is equal to unity inside the solid and zero outside. The step function of the solid is used for volume integrations needed for the analysis. The traditional weak form for Poisson's equation is modified by using this solution structure to eliminate the surface integration terms. The accuracy and implementation of the present mesh free method is illustrated for two-dimensional heat conduction problems governed by Poisson's equation. Satisfactory results are obtained when compared with analytical results and results from commercial finite element software.

Original language | English (US) |
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Title of host publication | Proceedings of the ASME Heat Transfer Division 2005 |

Pages | 157-165 |

Number of pages | 9 |

Edition | 1 |

DOIs | |

State | Published - Dec 1 2005 |

Event | 2005 ASME International Mechanical Engineering Congress and Exposition, IMECE 2005 - Orlando, FL, United States Duration: Nov 5 2005 → Nov 11 2005 |

### Publication series

Name | American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD |
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Number | 1 |

Volume | 376 HTD |

ISSN (Print) | 0272-5673 |

### Conference

Conference | 2005 ASME International Mechanical Engineering Congress and Exposition, IMECE 2005 |
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Country | United States |

City | Orlando, FL |

Period | 11/5/05 → 11/11/05 |

### Fingerprint

### Keywords

- Boolean operation
- Implicit solid element
- Meshfree method
- Solution structure
- Step function
- Uniform grid

### ASJC Scopus subject areas

- Mechanical Engineering
- Fluid Flow and Transfer Processes

### Cite this

*Proceedings of the ASME Heat Transfer Division 2005*(1 ed., pp. 157-165). (American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD; Vol. 376 HTD, No. 1). https://doi.org/10.1115/IMECE2005-80872

**Solution structures for imposing boundary conditions in mesh free analysis of heat conduction problems.** / Gu, Linxia; Kumar, Ashok V.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the ASME Heat Transfer Division 2005.*1 edn, American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD, no. 1, vol. 376 HTD, pp. 157-165, 2005 ASME International Mechanical Engineering Congress and Exposition, IMECE 2005, Orlando, FL, United States, 11/5/05. https://doi.org/10.1115/IMECE2005-80872

}

TY - GEN

T1 - Solution structures for imposing boundary conditions in mesh free analysis of heat conduction problems

AU - Gu, Linxia

AU - Kumar, Ashok V.

PY - 2005/12/1

Y1 - 2005/12/1

N2 - One of the main advantages of meshless methods is that it eliminates the mesh generation, but it is still necessary to place nodes with controlled spacing variation on the boundary and within the domain. However, due to lack of connectivity between nodes it is more difficult to interpolate the field variables and impose boundary conditions. In this paper, a mesh free method is presented for analysis using a structured grid that does not conform to the geometry of the domain. The geometry of the domain is independent of the structured grid and is represented using implicit equations. The implicit equations of the boundaries can be used to construct solution structures that satisfy boundary conditions exactly even though the nodes of the grid are not on the boundaries of the domain. The solution structures are constructed using the implicit equations of the boundary together with a piece-wise interpolation over the structured grid. The implicit equations are also used to construct step function of solid such that its value is equal to unity inside the solid and zero outside. The step function of the solid is used for volume integrations needed for the analysis. The traditional weak form for Poisson's equation is modified by using this solution structure to eliminate the surface integration terms. The accuracy and implementation of the present mesh free method is illustrated for two-dimensional heat conduction problems governed by Poisson's equation. Satisfactory results are obtained when compared with analytical results and results from commercial finite element software.

AB - One of the main advantages of meshless methods is that it eliminates the mesh generation, but it is still necessary to place nodes with controlled spacing variation on the boundary and within the domain. However, due to lack of connectivity between nodes it is more difficult to interpolate the field variables and impose boundary conditions. In this paper, a mesh free method is presented for analysis using a structured grid that does not conform to the geometry of the domain. The geometry of the domain is independent of the structured grid and is represented using implicit equations. The implicit equations of the boundaries can be used to construct solution structures that satisfy boundary conditions exactly even though the nodes of the grid are not on the boundaries of the domain. The solution structures are constructed using the implicit equations of the boundary together with a piece-wise interpolation over the structured grid. The implicit equations are also used to construct step function of solid such that its value is equal to unity inside the solid and zero outside. The step function of the solid is used for volume integrations needed for the analysis. The traditional weak form for Poisson's equation is modified by using this solution structure to eliminate the surface integration terms. The accuracy and implementation of the present mesh free method is illustrated for two-dimensional heat conduction problems governed by Poisson's equation. Satisfactory results are obtained when compared with analytical results and results from commercial finite element software.

KW - Boolean operation

KW - Implicit solid element

KW - Meshfree method

KW - Solution structure

KW - Step function

KW - Uniform grid

UR - http://www.scopus.com/inward/record.url?scp=33645674592&partnerID=8YFLogxK

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U2 - 10.1115/IMECE2005-80872

DO - 10.1115/IMECE2005-80872

M3 - Conference contribution

AN - SCOPUS:33645674592

SN - 0791842215

SN - 9780791842218

T3 - American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD

SP - 157

EP - 165

BT - Proceedings of the ASME Heat Transfer Division 2005

ER -