A dual mortar approach for mesh tying within a variational multiscale method for incompressible flow

Abstract

SUMMARY In the present work, the coupling of computational subdomains with non‐conforming discretizations is addressed in the context of residual‐based variational multiscale finite element methods for incompressible fluid flow. A mortar method using dual Lagrange multipliers is introduced for handling the coupling conditions at arbitrary fluid–fluid interfaces. Recently, mortar methods have been successfully applied in the field of nonlinear solid mechanics, for example, to weakly impose interface constraints for finite deformation contact. The focus of this study is on both the integration of the dual mortar approach into an existing variational multiscale finite element framework and the investigation of the resulting interplay between variational multiscale and coupling terms. We analyze the effects of either constraining only velocity or both velocity and pressure degrees of freedom at the internal interfaces using dual Lagrange multipliers. In analogy to other problem classes, we exploit the fact that the dual mortar approach allows for an efficient condensation of the additional Lagrange multiplier degrees of freedom from the global system of equations. As a result, the typical but often undesirable saddle‐point structure of this system is completely removed. The proposed method is validated numerically for various three‐dimensional examples, including a complex patient‐specific aneurysm, and its accuracy and efficiency in comparison with standard conforming discretizations is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

“…The enforcement of the mass conservation (Figure 8(g)) is optimally convergent in the transient convection-dominated flow regime independent of the embedded mesh location, which validates the convective and transient scalings in the interface terms in (17) and the applied upwinding scheme used to control the convective mass transport across the interface. Moreover, the optimal convergence in all norms demonstrates the right scalings for all fluid and ghost-penalty stabilization terms in (7) according to the different flow regimes. Moreover, the obtained convergence rates are in perfect agreement with [21].…”

“…For integrating terms on fluid volume-cells of the cut background elements, the recent developed technique from [29], which is based on the divergence theorem, has been applied. It should be noted that in (7), the interface conditions (5) and (6) at ff have not been included yet. The weak enforcement of the coupling conditions will be presented in Section 3.3.…”

“…All residual-based stabilization terms in (7) can be derived for each subdomain by decomposing velocity and pressure u i ; p i , as well as the corresponding weighting functions v i ; q i , in the weak formulation as…”

“…In contrast to a classical finite element discretization, as used for the embedded mesh T e , where the aforementioned residual-based stabilization terms sufficiently control possible instabilities, additional measures have to be taken on cut elements of the background mesh T b . As proposed in [25] and the references therein, in the interface region, the following additional face-oriented fluid and ghost-penalty stabilization terms from (7) are evaluated along inter-element faces of the background mesh T b :…”

A monolithic, mortar‐based interface coupling and solution scheme for finite element simulations of lithium‐ion cells

“…To circumvent this problem, we subdivide the entire integration domain into contiguous nonoverlapping integration cells, each carrying a smooth integrable portion of the overall integrand, as depicted in Figure . This procedure is called segment‐based integration and is well known from computational contact mechanics,() fluid mechanics, and mesh tying in solid mechanics . We adopt this integration technique to achieve maximum accuracy with our mortar‐based coupling scheme, as demonstrated in our numerical examples in Section 5.…”