##
**Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM).**
*(English)*
Zbl 1158.74532

Summary: It is well known that the displacement-based fully compatible finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems. It is, however, much more difficult to bound the solution from above for general problems in elasticity, and it has been a dream of many decades to find a systematical way to obtain such an upper bound. This paper presents a very important and unique property of the linearly conforming point interpolation method (LC-PIM): it provides a general means to obtain an upper bound solution in energy norm for elasticity problems. This paper conducts first a thorough theoretical studyon the LC-PIM: we derive its weak form based on variational principles, study a number of properties of the LC-PIM, and prove that LC-PIM is variationally consistent and that it produces upper bound solutions. We then demonstrate these properties through intensive numerical studies with many examples of 1D, 2D, and 3D problems. Using the LC-PIM together with the FEM, we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems, as shown in these example problems.

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74S05 | Finite element methods applied to problems in solid mechanics |

74G65 | Energy minimization in equilibrium problems in solid mechanics |

74B05 | Classical linear elasticity |

### Keywords:

numerical methods; mesh-free methods; point interpolation method; nodal integration; solution bound; elasticity
PDF
BibTeX
XML
Cite

\textit{G. R. Liu} and \textit{G. Y. Zhang}, Int. J. Numer. Methods Eng. 74, No. 7, 1128--1161 (2008; Zbl 1158.74532)

Full Text:
DOI

### References:

[1] | Lucy, The Astronomical Journal 8 pp 1013– (1977) |

[2] | . Smoothed Particle Hydrodynamics–A Meshfree Practical Method. World Scientific: Singapore, 2003. · Zbl 1046.76001 |

[3] | Nayroles, Computational Mechanics 10 pp 307– (1992) |

[4] | Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) |

[5] | Liu, International Journal for Numerical Methods in Engineering 20 pp 1081– (1995) |

[6] | Atluri, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 |

[7] | Liu, International Journal for Numerical Methods in Engineering 50 pp 937– (2001) |

[8] | Wang, International Journal for Numerical Methods in Engineering 54 pp 1623– (2002) |

[9] | Meshfree methods: Moving Beyond the Finite Element Method. CRC Press: Boca Raton, U.S.A., 2002. |

[10] | . An Introduction to Meshfree Methods and Their Programming. Springer: Dordrecht, The Netherlands, 2005. |

[11] | Liu, Computational Mechanics 36 pp 421– (2005) |

[12] | Chen, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) |

[13] | Liu, International Journal of Computational Methods 3 pp 401– (2006) |

[14] | Liu, International Journal of Computational Methods 2 pp 645– (2005) |

[15] | Zhang, International Journal for Numerical Methods in Engineering (2007) |

[16] | . The Finite Element Method (5th edn). Butterworth Heinemann: Oxford, U.K., 2000. |

[17] | . The Finite Element Method: A Practical Course. Butterworth Heinemann: Oxford, 2003. · Zbl 1027.74001 |

[18] | Oliveira, International Journal of Solids and Structures 4 pp 929– (1968) |

[19] | Variational Principle in Elasticity and Applications. Scientific Press: Beijing, 1982. |

[20] | . Computational Inelasticity. Springer: New York, 1998. · Zbl 0934.74003 |

[21] | Liu, Computational Mechanics 39 pp 859– (2007) |

[22] | Liu, International Journal for Numerical Methods in Engineering 71 pp 902– (2007) |

[23] | . Theory of Elasticity (3rd edn). McGraw: New York, 1970. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.