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To discretize the virtual work relations, let where represents the interpolation functions over an element, , , , , and respectively represent the nodal values of , , , , and ; is the number of nodes in an element.
The discretized form of in (3.3-3) 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 current spatial configuration 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. Since the parametric space is invariant, time derivatives are evaluated in a material frame. For example, the time derivative appearing in (3.3-3) becomes when evaluated at the parametric coordinates of the th integration point. All time derivatives are discretized using a backward difference scheme.
Similarly, the discretized form of may be written as where the terms in the first column are the discretized form of the linearization along : where The terms in the second column of the stiffness matrix in (3.3.3-4) are the discretized form of the linearization along : where The terms in the third column of the stiffness matrix in (3.3.3-4) are the discretized form of the linearization along : where and The discretization of in (3.4.4-3) has the form where . The summation is performed over all surface elements on which these boundary conditions are prescribed. The discretization of has the form where In this expression, is the antisymmetric tensor whose dual vector is (such that for any vector .