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Let where represents the interpolation functions over an element, respectively represent nodal values of , , and , and is the number of nodes in an element. Then the discretized form of in Eq.(3.2.1-3) may be written as where is the number of elements in , is the number of integration points in the th element, is the quadrature weight associated with the th integration point, and is the Jacobian of the transformation from the spatial frame to the parametric space of the element. In the above expression, and it is understood that , and are evaluated at the parametric coordinates of the th integration point.
Similarly, the discretized form of in Eqs.(3.2.1-5) and (3.2.1-9) may be written as where and is a discrete increment in time. In a numerical implementation, it has been found that evaluating from , where , yields more accurate solutions than evaluating it from the trace of .
For the various types of contributions to the external virtual work, a similar discretization produces and where In this case, represents the number of nodes on an element face. For a prescribed normal traction as given in (3.2.1-10)-(3.2.1-11), where is the skew-symmetric tensor whose dual vector is and is the third-order permutation pseudo-tensor. For a prescribed traction as given in (3.2.1-14)-(3.2.1-15), For a prescribed normal fluid flux as given in (3.2.1-16)-(3.2.1-17),