  We present a unified framework to tie overlapping meshes in solid mechanics
applications. This framework is a combination of the X-FEM method and the
mortar method, which uses Lagrange multipliers to fulfill the tying
constraints. As known, mixed formulations are prone to mesh locking which
manifests itself by the emergence of spurious oscillations in the vicinity of
the tying interface. To overcome this inherent difficulty, we suggest a new
coarse-grained interpolation of Lagrange multipliers. This technique consists
in selective assignment of Lagrange multipliers on nodes of the mortar side and
in non-local interpolation of the associated traction field. The optimal choice
of the coarse-graining spacing is guided solely by the mesh-density contrast
between the mesh of the mortar side and the number of blending elements of the
host mesh. The method is tested on two patch tests (compression and bending)
for different interpolations and element types as well as for different
material and mesh contrasts. The optimal mesh convergence and removal of
spurious oscillations is also demonstrated on the Eshelby inclusion problem for
high contrasts of inclusion/matrix materials. Few additional examples confirm
the performance of the elaborated framework.
