Theory Manual Version 3.6
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Subsubsection 7.1.9.11: Frictional Terms Up Subsection 7.1.9: Sliding-Elastic Section 7.2: Biphasic Contact

#### 7.1.9.12 Integration Scheme

In this formulation, a Gaussian quadrature integration scheme is adopted. The general form of the contact integral (e.g. eq.(7.1.9.9-2) or eq.(7.1.9.9-8)) may be integrated numerically as where is the number of element faces on , is the number of integration points on the element face of , is the weight associated with the integration point, and where it should be understood that terms associated with (such as , , etc.) are evaluated at the parametric coordinates , associated with the integration point, and terms associated with (such as ) are evaluated at the parametric coordinates of contact or , defined by Eqs.(7.1.9.2-1) and (7.1.9.1-1) for cases of stick and slip, respectively. The subscript appearing in the terms associated with has been added to emphasize that there may be up to distinct element faces on associated with all the integration points on the element face of , based on the location of the contact point on as defined by either eq.(7.1.9.2-1) or (7.1.9.1-1).
In a similar fashion, the contact stiffness may be integrated numerically as where the matrix of tensors is a general representation of the stiffness terms given explicitly in either eq.(7.1.9.9-6) or Eqs.(7.1.9.10-3) and (7.1.9.11-3). In this expression, is the vector of incremental changes in the degrees of freedom of the node of the element face on , and the node of the element face on which contains the contact point associated with the integration point on the element face of ; the specific form of will be dictated by the stick/slip status. A more detailed treatment of Gaussian quadrature, and a discussion of the benefits of this scheme versus nodal integration, may be found in our previous work [8].
Subsubsection 7.1.9.11: Frictional Terms Up Subsection 7.1.9: Sliding-Elastic Section 7.2: Biphasic Contact