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Title: On G space theory
Authors: Liu, G.R. 
Keywords: Compatibility
Finite element method
G space
Generalized smoothed Galerkin
Meshfree method
Numerical method
Point interpolation method
Solution bound
Variational principle
Weakened weakform
Issue Date: 2009
Citation: Liu, G.R. (2009). On G space theory. International Journal of Computational Methods 6 (2) : 257-289. ScholarBank@NUS Repository.
Abstract: This paper presents an examinations of the G space theory that was established recently for a unified formulation of compatible and incompatible models for mechanics problems using the finite element and meshfree settings. Using the generalized gradient smoothing technique, we first give a general definition for G spaces with more details on the G1 space containing both continuous and discontinuous functions. The physical meanings and implications of various numerical treatments used in the G space theory are discussed in detail. Both normed and un-normed G spaces are discussed with emphases on the normed G spaces. Some important properties and a set of useful inequalities for the normed G spaces are proven in theory and analyzed in detail. Because discontinuous functions are allowed in a G space, much more types of function approximation methods and techniques can be used to create shape functions for numerical models. These models can be compatible and incomputable but are all stable and converge to the exact solution to the corresponding strong formulation as long as it is well-posed, based on the normed G space theory. Methods based on normed G space theory does not use the derivatives of the displacement functions in the formulation and is known as the weakened weak (W 2) formulation that has a number of attractive properties such as conformability, softness, upper/lower bound, superconvergence, and ultra accuracy. © World Scientific Publishing Company.
Source Title: International Journal of Computational Methods
ISSN: 02198762
DOI: 10.1142/S0219876209001863
Appears in Collections:Staff Publications

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