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Isogeometric mortar methods
Ericka Brivadis, Annalisa Buffa, Barbara Wohlmuth, Linus Wunderlich
TL;DR
Multipatch isogeometric analysis needs flexible interface coupling, but the Lagrange multiplier space must satisfy approximation and inf-sup stability requirements. The paper analyzes spline multiplier choices theoretically and numerically, showing stable alternatives with different degree pairings and cross-point requirements.
Problem
Multipatch isogeometric analysis requires interface coupling, while selecting a Lagrange multiplier space must balance approximation order with inf-sup stability.
Method
The paper investigates spline multiplier spaces with degrees p, p−1, and p−2 relative to a primal degree p, combining theoretical analysis with numerical examples.
Results
The p/p−1 pairing is unstable, whereas the other pairings satisfy the stability condition; equal-order pairing achieves stability through a boundary modification.
Takeaways & Limitations
The study presents two uniformly stable spline-space choices: equal order with local boundary degree reduction, or multiplier degree p−2 without cross-point modification.
Takeaways & Limitations
The analysis assumes globally quasi-uniform meshes, excluding anisotropic and graded meshes, and boundary modification adds implementation effort.
Abstract
from arXiv · showhide
The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf-sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects. Keywords: isogeometric analysis, mortar methods, inf-sup stability, cross point modification.
1. Introduction
The paper studies isogeometric mortar coupling for multipatch domains, focusing on selecting Lagrange multiplier spline spaces that satisfy approximation and inf-sup stability requirements.
- Isogeometric analysis represents geometry and PDE approximations with B-Splines or NURBS.
- Multipatch discretizations require coupling techniques across interfaces between subdomains.
- Weak coupling is preferred because it retains mesh flexibility at interfaces, motivating mortar methods for domain decomposition.
- The central design issue is choosing a Lagrange multiplier space satisfying sufficient approximation order and inf-sup stability.
- For primal spline degree p, the paper investigates multiplier degrees p, p −1, and p −2, which have different characteristic features.
- The article defines isogeometric mortar methods, details three multiplier types, and examines theoretical results numerically with additional examples.
452. B-Splines and NURBS basics
This section introduces B-Splines and NURBS, their tensor-product construction, knot-based meshes, refinement, parametrizations, and the regularity assumptions used for analysis.
- Univariate B-Splines: B-Splines are piecewise positive polynomials of degree p with local support over at most p + 1 elements.
- Univariate B-Splines: Knot multiplicity determines inter-element continuity, with continuity C^(p−m_j) at breakpoint ζ_j.
- Refinement: The paper considers h-refinement, which inserts knots while keeping the spline degree fixed.
- Multivariate spaces: Multivariate B-Splines are constructed by tensorizing univariate B-Splines across parametric directions.
- NURBS: NURBS are rational functions of multivariate B-Splines defined using positive weights, while unit weights recover B-Splines.
- Parametrization: The geometric mapping F sends the parametric domain to the physical domain and transfers the parametric mesh to a physical mesh.
- Analysis assumptions: The analysis assumes a smooth, bi-Lipschitz parametrization and globally quasi-uniform meshes, excluding anisotropic and graded meshes.
- Analysis assumptions: The quasi-uniformity assumption is made to reduce proof technicality, and milder mesh assumptions may yield the same results.
3. Isogeometric mortar methods
The paper formulates isogeometric mortar methods for multipatch elliptic problems using weak interface continuity and discrete Lagrange multiplier spaces built on slave meshes.
- Computational domain: The framework allows geometrically conforming and slave conforming interfaces, with master and slave sides assigned for each interface.
- Mortar formulation: The discrete mortar method uses broken Sobolev spaces and weakly enforces continuity through interface Lagrange multipliers.
- Mortar formulation: The multiplier space is assembled as a product of interface spaces built on slave meshes, and the discrete problem is posed as a saddle point system.
- Mortar formulation: The Lagrange multiplier approximates the normal flux across the skeleton.
- Stability and approximation: Well-posedness and optimal convergence require uniform inf-sup stability and adequate approximation order for the multiplier space.
- Convergence: Under these assumptions, the primal solution has an a-priori convergence estimate governed by solution regularity, primal degree, and multiplier approximation order.
- Convergence: The mesh-size ratio between master and slave sides can enter estimates in nonconforming or three-dimensional cases, but global quasi-uniformity removes its role here.
- Convergence: Choosing η(l) = p_s(l) −1/2 yields optimal mortar-method convergence, while additional regularity can improve the dual estimate.
4. Possible choices of Lagrange multiplier spaces
The paper examines spline-based Lagrange multiplier spaces for isogeometric mortar methods, comparing their approximation and inf-sup stability properties. Three pairings are assessed: p/p−1, p/p−2, and p/p with boundary modification.
- Additional remarks: Biorthogonal multiplier bases can enable local elimination of coupling degrees of freedom, but constructing them with the desired approximation properties is nontrivial for higher-order cases.The paper therefore does not pursue this approach in the subsequent analysis.
- Choice 1: unstable pairing p/p−1: The p/p−1 instability is associated with a checkerboard mode and can be recovered numerically using a staggered grid or a coarse dual mesh.The checkerboard construction applies to uniform knot vectors and tensor-product extensions.
- Choice 2: stable pairing p/p−2: The p/p−2 pairing satisfies both the inf-sup and approximation assumptions, yielding an order p−1/2 convergence rate.Its stability proof uses derivative and integral mappings together with an auxiliary stability result for degree p−1 splines.
4005. Numerical results
Numerical tests evaluate inf-sup stability and convergence across spline pairings, boundary conditions, singularities, nonmatching interfaces, and elasticity problems. The results support the theoretical stability and optimality claims while identifying effects of solution regularity and geometry approximation.
- Inf-sup evaluation: The numerical study evaluates inf-sup constants for the proposed spaces and additional lower-degree choices, using the Chapelle–Bathe technique.The tests consider primal spaces with and without homogeneous Dirichlet conditions and examine h- and p-dependence.
- Inf-sup evaluation: The inf-sup condition is satisfied for pairings whose primal and dual degrees have the same parity.For primal spaces without boundary conditions, p-dependence is reasonable; with boundary conditions, exponential behavior is observed.
- Scalar problems: For equal-order pairings, the L2 error converges with order p + 1, and asymptotically reaches the optimal order at each refinement step.The reported comparison between matching and nonmatching meshes shows no significant quantitative difference, although it uses similar meshes rather than identical control-point repartitions.
- Singular scalar problem: The re-entrant-corner example confirms optimality relative to solution regularity, while the dual variable can converge at only 1/6 because of interface regularity.The singularity also produces an L2 pollution effect across subdomains, whereas subdomain-wise H1 rates can improve away from the singularity.
- Additional examples: Across degree pairings, lowering the dual degree does not deteriorate primal accuracy, and the methods achieve best-approximation rates; linear elasticity behaves similarly to scalar problems.For the nonmatching-interface example, the method is robust, while the additional geometry error has a quite small influence.
6. Conclusion
The paper investigates isogeometric mortar formulations mathematically and practically, comparing spline multiplier spaces of degrees p, p−1, and p−2. Equal-order pairing provides optimal results with boundary modification, while p/p−2 avoids that modification but may reduce convergence order.
- The study considers dual spline spaces of degree p, p−1, and p−2 for a given primal order p.
- The p/p−1 pairing is unstable, whereas the other pairings satisfy the inf-sup condition.
- Equal-order pairing guarantees optimal results, but its stability requires a boundary modification.
- Numerical examples show that the mortar method handles geometry-approximation difficulties and singularities, with several cases attaining superior convergence orders.
- Spaces tailored to contact problems remain an ongoing research subject.