A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier

Abstract

The mortar nite element method allows the coupling of di erent discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modi ed trace space which serves as a space of Lagrange multipliers. In this paper, this Lagrange multiplier space is replaced by a dual space without losing the optimality of the method. The advantage of this new approach is that the matching condition is much easier to realize. In particular, all the basis functions of the new method are supported in a few elements. The mortar map can be represented by a diagonal matrix; in the standard mortar method a linear system of equations must be solved. The problem is considered in a positive de nite nonconforming variational as well as an equivalent saddle-point formulation.

Isogeometric dual mortar methods for computational contact mechanics

“…Using these methods based on a standard Lagrange multiplier interpolation, a system of increased size containing both displacement and Lagrange multiplier degrees of freedom has to be solved. In this work, we follow a different approach using dual shape functions for the Lagrange multiplier, which were initially introduced in domain decomposition problems [19,45] and extended to contact problems in [20][21][22][23][24][25][26][27] and recently reviewed in [29]. While dual mortar methods are meanwhile well-established in finite elements, the present work, to the authors knowledge, is the first application of dual basis functions in the context of IGA for both domain decomposition and finite deformation frictional contact.…”

“…While dual mortar methods are meanwhile well-established in finite elements, the present work, to the authors knowledge, is the first application of dual basis functions in the context of IGA for both domain decomposition and finite deformation frictional contact. Dual basis functions are characterized by fulfilling a biorthogonality condition [19]  γ (1) c,h…”

“…Proposing an alternative approximation of the Lagrange multiplier, so-called dual mortar methods were originally introduced in the context of domain decomposition [19] and later extended to small deformation frictionless contact in 2D [20] and 3D [21], small deformation frictional contact [22], finite deformation frictionless contact [23,24] and finite deformation frictional contact [25]. Since NURBS are naturally associated with higher order approximations, the extension of the dual mortar method to second order Lagrange finite elements in [26,27] including optimal a priori error estimates is also worth mentioning.…”

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

“…On this mesh the finite dimensional space M lc δm ⊂ L 2 (Γ m ) is defined, and we also set M lc δ = m∈M M lc δm . In the present context the space M lc δm contains piecewise (discontinuous) polynomials of degree k λ in the interior of the trace and the polynomial functions of degree k λ − 1 on the first and last intervals of the discretization, [11,33,8]. The discrete version of problem (3) within this framework is:…”

Symmetrization of Various Friction Models Based on an Augmented Lagrangian Approach

“…In addition, good results for the patch-test have been shown also in [3] exploiting the, so-called, segment-to-segment approach, coinciding with the Mortar method with penalty descriptions of the contact traction. Socalled dual Lagrange multipliers have been developed for non-frictional contact in Wohlmuth [7] allowing to condense degrees of freedom for contact traction. Using this approach, the frictional constraints should be carefully treated as a sequence of the Tresca type friction model, see [8].…”