Please use this identifier to cite or link to this item: https://doi.org/10.1002/nme.2719
Title: A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory
Authors: Liu, G.R. 
Keywords: Compatibility
Finite element method
G space
Galerkin weak form
Mesh-free method
Numerical method
Point interpolation method
Solution bound
Variational principle
Weakened weak form
Issue Date: 26-Feb-2010
Citation: Liu, G.R. (2010-02-26). A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory. International Journal for Numerical Methods in Engineering 81 (9) : 1093-1126. ScholarBank@NUS Repository. https://doi.org/10.1002/nme.2719
Abstract: This paper introduces a G space theory and a weakened weak form (W2) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W2 formulation works for both finite element method settings and mesh-free settings, and W2 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 W2 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 W2 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. © 2009 John Wiley & Sons, Ltd.
Source Title: International Journal for Numerical Methods in Engineering
URI: http://scholarbank.nus.edu.sg/handle/10635/84783
ISSN: 00295981
DOI: 10.1002/nme.2719
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