Theory Manual Version 3.6
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Subsubsection 7.1.9.8: Linearization Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.10: Frictionless Terms

#### 7.1.9.9 Discretization

Let the continuous variables on the primary and secondary surfaces be interpolated over each element face according to 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.
Stick
Applying the discretization to eq.(7.1.9-5), the contact integral becomes where and is defined by eq.(7.1.9.5-1) in the penalty case and eq.(7.1.9.6-1) when augmented Lagrangian regularization is used. Individual terms may now be discretized and placed into matrix notation, facilitating their substitution into eq.(7.1.9.8-4). A straightforward application of eq.(7.1.9.9-1) to eq.(7.1.9.8-2) yields where and is the vector of incremental changes in the degrees of freedom to the node associated with the element face on , and to the node of the element face on associated with the current integration point on . Furthermore, discretizing eq.(7.1.9.8-3) produces Substituting eq.(7.1.9.9-3) and eq.(7.1.9.9-5) into eq.(7.1.9.8-4) and applying the discretization of eq.(7.1.9.9-1) yields the directional derivative of the virtual work in stick as An examination of eq.(7.1.9.9-6) shows that the stiffness matrix associated with this contact formulation is not symmetric in stick.
Although the stiffness matrix of eq.(7.1.9.9-6) can be cast in a more traditional form, as was done in [12], splitting up like terms is a more natural way to implement the final equations, and provides some insight into the resulting matrix structure. A discussion of how to numerically evaluate the integrals in the above equations is deferred until Section 7.1.9.12↓.
Slip
In the case of slip, the contact integral of eq.(7.1.9-5) can be split into normal and tangential parts, , such that Discretizing this expression yields where and is given by eq.(7.1.9.5-2) in the penalty formulation and eq.(7.1.9.6-2) with augmented Lagrangian regularization.
For the following linearization and discretization the normal and tangential components, representing frictionless and frictional contributions to the contact integral, will be treated separately and may then be added together as in eq.(7.1.9.9-7).
Subsubsection 7.1.9.8: Linearization Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.10: Frictionless Terms