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Weakened weak form
Weakened weak form (or W2 form) 〔G.R. Liu. "A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems". ''International Journal for Numerical Methods in Engineering'', 81: 1093–1126, 2010〕 is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems.
==Description==
For simplicity we choose elasticity problems (2nd order PDE) for our discussion.〔Liu, G.R. 2nd edn: 2009 ''Mesh Free Methods'', CRC Press. 978-1-4200-8209-9〕 Our discussion is also most convenient in reference to the well-known weak and strong form. In a strong formulation for an approximate solution, we need to assume displacement functions that are 2nd order differentiable. In a weak formulation, we create linear and bilinear forms and then search for a particular function (an approximate solution) that satisfy the weak statement. The bilinear form uses gradient of the functions that has only 1st order differentiation. Therefore, the requirement on the continuity of assumed displacement functions is weaker than in the strong formulation. In a discrete form (such as the Finite element method, or FEM), a sufficient requirement for an assumed displacement function is piecewise continuous over the entire problems domain. This allows us to construct the function using elements (but making sure it is continuous a long all element interfaces), leading to the powerful FEM.
Now, in a weakened weak (W2) formulation, we further reduce the requirement. We form a bilinear form using only the assumed function (not even the gradient). This is done by using the so-called generalized gradient smoothing technique,〔Liu GR, "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'' Vol.5 Issue: 2, 199–236, 2008〕 with which one can approximate the gradient of displacement functions for certain class of discontinuous functions, as long as they are in a proper G space.〔Liu GR, "On G Space Theory", ''International Journal of Computational Methods'', Vol. 6 Issue: 2, 257–289, 2009〕 Since we do not have to actually perform even the 1st differentiation to the assumed displacement functions, the requirement on the consistence of the functions are further reduced, and hence the weakened weak or W2 formulation.
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