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**A G space theory and a weakened weak (\(W^2\)) form for a unified formulation of compatible and incompatible methods. I: Theory.**
*(English)*
Zbl 1183.74358

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

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