Galerkin’s weighted residual method

Next, we apply Galerkin s weighted residual method and reduce the order of integration of the various terms in the above equations using the Green-Gauss Theorem (9.1.2) for each element. For a simpler presentation we will deal with each term in the above equations separately. The terms of the x-component (eqn. (9.95)) of the penalty formulation momentum balance become... 

The unknown nodal displacements are obtained using Galerkin s weighted residual method. The inner product of the governing equation with respect to each of the interpolation functions is set to zero over the whole domain 2. However, the 4 order derivative term in governing equation requires the interpolation function to have continuity. In other words, the first derivatives of Nj with respect to x and z should be continuous along the inter-element boundary to avoid infinity in the integration of the so-called "stiffness" matrix. Hence, we introduce a new variable O such that... 

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