Theory Manual Version 3.6
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Subsubsection 7.2.9.1: Linearization Up Subsection 7.2.9: Biphasic Stick Subsection 7.2.10: Biphasic Slip

#### 7.2.9.2 Discretization

Let the continuous variables on the primary and secondary surfaces be interpolated over element faces according to [8, 114] where represent interpolation functions on the element faces of , is the number of nodes and interpolation functions on each primary element face, is the number of nodes and interpolation functions on the secondary element face which is intersected by the ray issued from the integration point on the primary element face, and , ,, and represent respective nodal values of , , , and . From this point forward the summation signs will be written simply as , where it is assumed they have the same meaning described above.
Inserting the discretization of eq.(7.2.9.2-1) into eq.(7.2.1-4) yields where the residuals are and is obtained from eq.(7.2.4-1) in the penalty case and from eq.(7.2.5-1) if augmented Lagrangian regularization is employed. Similarly, is calculated from either eq.(7.2.4-4) or eq.(7.2.5-4) for penalty and augmented Lagrange methods, respectively.
We may now discretize individual terms and place them into matrix notation, anticipating their substitution into eq.(7.2.9.1-4). By placing eq.(7.2.9.2-2) into Eqs.(7.2.9.1-2)-(7.2.9.1-3) and inserting the resulting linearizations into eq.(7.2.9.1-4), we obtain the stiffness matrix for biphasic stick as where and In eq.(7.2.9.2-3), is the vector of incremental changes in the degrees of freedom of the th node of the current element face on . Similarly, represents the incremental changes in the degrees of freedom of the th node of the element face on which contains the intersection point associated with the th integration point on the current element face on [12, 114]. See Section 7.1.9.12↑ for a description of the Gaussian quadrature integration scheme used to evaluate residuals and stiffness matrices (e.g. Eqs.(7.2.9.2-2) and (7.2.9.2-3)). Briefly, it should be noted that for all terms associated with (e.g. , , ) there may be distinct element faces on associated with all the integration points on the th element face of , based on the location of obtained from eq.(7.2.3.2-1).
In Section 7.1.9↑ on sliding-elastic frictional contact, we split the contact stiffness matrices in a different way, which allowed us to clearly separate like terms. Here, due to the complexity of the biphasic contact formulation, it was determined that the form of eq.(7.2.9.2-3) provides more clarity.
Subsubsection 7.2.9.1: Linearization Up Subsection 7.2.9: Biphasic Stick Subsection 7.2.10: Biphasic Slip