Transformed dual basis for higher-order mortar

A dual (biorthogonal) Lagrange multiplier basis satisfies

where is the standard basis on the secondary face . Mortar mechanical contact divides the weighted gap of node by , so every must be positive. On QUAD8 and TRI6 faces, the faces of HEX20 and TET10 elements, it is not: at QUAD8 vertices and at TRI6 vertices, where the dual shape functions vanish (Popp et al. (2012), Eqs. (4.3) and (4.4)).

libMesh therefore builds the dual basis of a SECOND order Lagrange multiplier on these faces against the locally quadratic transformed basis of Popp et al. (2012), Sec. 4.4.1, defined in terms of the standard basis as

where is a vertex node and are its adjacent mid-edge nodes . The transform keeps the partition of unity and makes all weights positive:

The transform is applied automatically whenever such a multiplier sets use_dual = true, which is the default for mortar in the Contact action. Other faces, including QUAD9, and FIRST order multipliers keep the standard dual basis. Because biorthogonality holds against , the coupling matrix is not diagonal on transformed faces.

The Petrov-Galerkin approach (Petrov-Galerkin approach for Lagrange multipliers) weights the gap with the standard basis , whose nodal integrals are not positive on these faces. MOOSE therefore reports an error when use_petrov_galerkin = true is combined with a SECOND order dual multiplier on QUAD8 or TRI6 secondary faces.