A G space theory and a weakened weak (\(W^2\)) form for a unified formulation of compatible and incompatible methods. I: Theory.

*(English)*Zbl 1183.74358Summary: This paper introduces a G space theory and a weakened weak form \((W^{2})\) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The \(W^{2}\) formulation works for both finite element method settings and mesh-free settings, and \(W^{2}\) models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for \(W^{2}\) formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the \(W^{2}\) formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ‘close-to-exact’ stiffness, upper bounds and ultra accuracy.

##### MSC:

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

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

##### Keywords:

numerical method; mesh-free method; G space; weakened weak form; point interpolation method; finite element method; solution bound; variational principle; Galerkin weak form; numerical method; compatibility
PDF
BibTeX
XML
Cite

\textit{G. R. Liu}, Int. J. Numer. Methods Eng. 81, No. 9, 1093--1126 (2010; Zbl 1183.74358)

Full Text:
DOI

##### References:

[1] | Zienkiewicz, The Finite Element Method (2000) · Zbl 0962.76056 |

[2] | Liu, Finite Element Method: A Practical Course (2003) |

[3] | Oliveira Eduardo, Theoretical foundations of the finite element method, International Journal of Solids and Structures 4 pp 929– (1968) |

[4] | Liu, Smoothed Particle Hydrodynamics-A Mesh Free Particle Method (2003) · Zbl 1046.76001 |

[5] | Liu, An Introduction to MFree Methods and their Programming (2005) |

[6] | Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2009) |

[7] | Pian, Hybrid and Incompatible Finite Element Methods (2006) · Zbl 1110.65003 |

[8] | Quarteroni, Numerical Approximation of Partial Differential Equations (1994) |

[9] | Liu, A smoothed finite element method for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047 |

[10] | Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 pp 902– (2007) · Zbl 1194.74432 |

[11] | Dai, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elements in Analysis and Design 43 pp 847– (2007) |

[12] | Eringen, On nonlocal elasticity, International Journal of Engineering Sciences 10 pp 233– (1972) · Zbl 0247.73005 |

[13] | Zhang, Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure, Physics Letters A 349 pp 370– (2006) |

[14] | Lucy, Numerical approach to testing the fission hypothesis, Astronomical Journal 82 pp 1013– (1977) |

[15] | Liu, An overview on smoothed particle hydrodynamics, International Journal of Computational Methods 5 pp 135– (2008) · Zbl 1257.76092 |

[16] | Monaghan, Why particle methods work (hydrodynamics), SIAM Journal on Scientific and Statistical Computing 3 pp 422– (1982) · Zbl 0498.76010 |

[17] | Chen, Regularization of material instabilities by mesh free approximations with intrinsic length scales, International Journal for Numerical Methods in Engineering 47 pp 1303– (2000) |

[18] | Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081 |

[19] | Liu, A linearly conforming point interpolation method (LC-PIM) for 2D mechanics problems, International Journal for Computational Methods 2 pp 645– (2005) · Zbl 1137.74303 |

[20] | Zhang, A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72 pp 1524– (2007) · Zbl 1194.74543 |

[21] | Liu, A normed G space and weakened weak (W2) formulation of a cell-based smoothed point interpolation method, International Journal of Computational Methods 6 (1) pp 147– (2009) |

[22] | Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, International Journal of Computational Methods 5 (2) pp 199– (2008) · Zbl 1222.74044 |

[23] | Liu, A point interpolation method for two-dimensional solids, International Journal for Numerical Methods in Engineering 50 pp 937– (2001) · Zbl 1050.74057 |

[24] | Wang, A point interpolation meshless method based on the radial basis functions, International Journal for Numerical Methods in Engineering 54 pp 1623– (2002) · Zbl 1098.74741 |

[25] | Liu, A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods 3 pp 401– (2006) · Zbl 1198.74120 |

[26] | Liu, Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM), International Journal for Numerical Methods in Engineering 74 pp 1128– (2008) · Zbl 1158.74532 |

[27] | Liu, A node-based smoothed finite element method (N-SFEM) for upper bound solutions to solid mechanics problems, Computers and Structures 87 pp 14– (2009) |

[28] | Liu, An edge-based smoothed finite element method (E-SFEM) for static and dynamic problems of solid mechanics, Journal of Sound and Vibration 320 pp 1100– (2009) |

[29] | Nguyen, A face-based smoothed finite element method (F-SFEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, International Journal for Numerical Methods in Engineering 78 pp 324– (2009) · Zbl 1183.74299 · doi:10.1002/nme.2491 |

[30] | Liu, Edge-based smoothed point interpolation method (ES-PIM), International Journal of Computational Methods 5 (4) pp 621– (2008) |

[31] | Peraire, Lecture Notes on Finite Element Methods for Elliptic Problems (1999) |

[32] | Hughes, The Finite Element Method (1987) · Zbl 0634.73056 |

[33] | Zhang, The upper bound property for solid mechanics of the linearly conforming radial point interpolation method (LC-RPIM), International Journal of Computational Methods 4 (3) pp 521– (2007) · Zbl 1198.74123 |

[34] | Strang, An Analysis of the Finite Element Method (1973) · Zbl 0356.65096 |

[35] | Liu, On a G space theory, International Journal of Computational Methods 6 (2) pp 257– (2009) |

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.