Frontiers in Mortar Methods for Isogeometric Analysis

Complex geometries as common in industrial applications consist of multiple patches, if spline based parametrizations are used. The requirements for the generation of analysis-suitable models are increasing dramatically since isogeometric analysis is directly based on the spline parametrization and nowadays used for the calculation of higher-order partial differential equations. The computational, or more general, the engineering analysis necessitates suitable coupling techniques between the different patches. Mortar methods have been successfully applied for coupling of patches and for contact mechanics in recent years to resolve the arising issues within the interface. We present here current achievements in the design of mortar technologies in isogeometric analysis within the Priority Program SPP 1748, Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.

💡 Research Summary

The paper provides a comprehensive overview of recent advances in mortar methods tailored for isogeometric analysis (IgA), focusing on the challenges posed by complex industrial geometries that are typically represented as multi‑patch spline models. Since IgA directly utilizes NURBS and B‑spline basis functions from CAD, it offers high‑order continuity and the ability to solve higher‑order partial differential equations. However, the decomposition of a domain into hundreds or thousands of patches creates non‑conforming interfaces that must be coupled efficiently and accurately.

The authors begin by recalling the origins of mortar techniques in the early 1990s, emphasizing their weak (integral) coupling nature as opposed to strong point‑wise constraints. By introducing Lagrange multipliers in a dual (bi‑orthogonal) form, mortar methods achieve variational consistency, satisfy inf‑sup stability, and enable the condensation of multiplier degrees of freedom via Schur complements. This theoretical foundation is then adapted to the IgA setting, where the high‑order spline spaces demand specially constructed multiplier spaces that preserve the smoothness and approximation properties of the primal fields.

A major portion of the manuscript is devoted to recent trends in IgA‑based mortar domain decomposition. The authors discuss:

1. Higher‑order patch coupling – Strategies for selecting multiplier spaces that match the polynomial degree and continuity of the underlying spline basis, ensuring optimal convergence rates and passing the patch test.
2. Multiphysics and multidimensional coupling – Extension of mortar transfer operators to couple 1D–3D, 2D–3D, and volume–surface interactions. Applications include fluid‑structure interaction (FSI), contact mechanics, and multi‑scale molecular dynamics (MD)–finite element (FE) coupling. In the MD‑FE context, a partition‑of‑unity weighting is attached to atoms, and an L²‑projection acts as a low‑pass filter, removing high‑frequency components that cannot be represented on the coarse FE mesh, thereby stabilizing the coupling.
3. Parallel implementation challenges – The assembly of mortar transfer operators requires integration over intersections of non‑matching meshes. For large‑scale simulations, a naïve global search is infeasible. The paper describes hierarchical k‑d tree searches combined with space‑filling curves to detect intersecting element pairs efficiently, distributing both intersection detection and quadrature evaluation across processors. The MoonoLith library is highlighted as a current implementation that supports fully parallel surface and volume mortar transfers.

The authors illustrate the versatility of the approach with several demanding applications. In cardiac simulations, they couple a finite‑difference Navier–Stokes solver for blood flow with an anisotropic fiber‑reinforced solid model, while simultaneously enforcing contact constraints on the prosthetic valve leaflets. The mortar formulation ensures a partition‑of‑unity multiplier space, eliminating leakage at the fluid‑structure interface and providing a stable implicit coupling. In geoscience, non‑conforming fracture network meshes are coupled via mortar operators to model flow through porous media. Multi‑scale mechanics examples demonstrate the MD‑FE coupling, where the mortar operator acts as a frequency filter, suppressing spurious wave reflections and achieving stable energy transfer.

Numerical experiments confirm that traditional node‑wise contact formulations fail the patch test and exhibit sub‑optimal convergence, whereas the variationally consistent mortar contact passes the test and attains the expected convergence order. The paper also discusses the impact of mortar condensation on system size, showing significant reductions in global degrees of freedom without sacrificing accuracy.

In the concluding section, the authors stress that mortar‑based IgA coupling simultaneously addresses three critical challenges: (i) maintaining high‑order continuity across patches, (ii) enabling flexible multi‑physics and multi‑scale interactions, and (iii) providing scalable parallel implementations for industrial‑scale problems. Future research directions include the design of even more efficient multiplier spaces, rigorous stability analysis for nonlinear and dynamic problems, and GPU‑accelerated mortar operators for real‑time simulations. Overall, the work positions mortar methods as a cornerstone technology for bringing complex CAD models directly into high‑fidelity, large‑scale computational mechanics workflows.