Theory Manual Version 3.6
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Subsection 7.2.7: Linearization and Discretization Outline Up Section 7.2: Biphasic Contact Subsection 7.2.9: Biphasic Stick

### 7.2.8 Definitions and Notation

Evaluating the linearizations of eq.(7.2.1-4) requires directional derivatives of kinematic quantities, some of which depend on the stick-slip status. To simplify the presentation, the continuum linearization is presented only for a few select quantities. The remainder of the linearization is deferred until after discretization, as many expressions are much easier to manipulate in discretized form. To keep the equations more manageable, the discretization of slip is split into frictionless and frictional terms, and the final form of the stiffness matrices follows this split. The full model is easily obtained by summing the frictionless and frictional contributions. Here we emphasize that, as a consequence of the double-loop Uzawa algorithm discussed in Section 7.2.5↑ (reprised from [114]), all Lagrange multipliers are updated outside of each Newton step, thus , where is any Lagrange multiplier, i.e. , etc. As a consequence, Lagrange multipliers do not appear in any of the linearized or discretized equations. In what follows, Greek indices are associated with covariant basis vectors on the contacting surfaces, and thus vary from 1 to 2. Repeated Greek indices indicate implicit summation over their range.
Many kinematic quantities remain unchanged from Section 7.1.9↑ on elastic frictional contact [114]. We thus accept without repetition the definitions of , , , , , , , , , , , , , , , , , , and . These terms are all defined in Sections 7.1.9.10↑-7.1.9.11↑ [114]. Directional derivatives which will not be duplicated here include , , , , , , , and (see Sections 7.1.9.8↑, 7.1.9.10↑ and 7.1.9.11↑).
The final residual vectors and stiffness matrices presented below are written in integral form. In the FEBio implementation, a Gaussian quadrature scheme is adopted to perform numerical integration. A detailed treatment of Gaussian quadrature, and equations for numerically integrating the contact integrals and stiffnesses, may be found in Section 7.1.9.12↑.
As a final implementation detail, we note that multiplying all biphasic entries in the residual vector and stiffness matrices by the time step leads to better convergence. In this context, biphasic refers to all terms which are not purely related to solid-solid contact. In the residual, this is all terms which are not (see below). Similarly, this includes all stiffness entries except the solid-solid terms (see below).
Subsection 7.2.7: Linearization and Discretization Outline Up Section 7.2: Biphasic Contact Subsection 7.2.9: Biphasic Stick