Bidirectional T-Patch Extraction in IgA

• Bidirectional T-patch extraction is a methodology for coupling non-matching multi-patch domains in IgA, enabling robust PDE solutions on complex geometries.
• It employs bidirectional information flow, artificial interfaces, and the fat vertex concept to enforce continuity at T-junctions.
• The approach enhances solver performance in IETI-DP frameworks while maintaining favorable condition numbers and stability in complex engineering simulations.

Bidirectional T-patch extraction is a class of methodologies designed to address the segmentation and coupling of non-matching multi-patch domains, specifically in the context of isogeometric analysis (IgA) where domain patches can meet at T-junctions. These methods emphasize bidirectional (mutual) information flow and coupling across irregular interfaces (T-junctions), moving beyond previous techniques that required conforming interfaces along whole edges. The incorporation of bidirectional T-patch extraction enables the robust and flexible solution of partial differential equations (PDEs) on complex domains by explicit handling of geometric singularities via advanced domain decomposition strategies.

1. Background: Multi-Patch Domains and T-Junctions

Isogeometric analysis often employs multi-patch representations to describe computational domains, using patches parameterized by, e.g., B-splines. In traditional approaches, patch interfaces are required to match fully along their boundaries; this ensures conformity of the global discretization and facilitates inter-patch continuity constraints. However, in practical engineering and geometric modeling scenarios—such as in CAD or moving component simulations—domains with T-junctions, where one patch abuts the interior of another's edge, are common. Such configurations lead to non-matching interfaces, introducing additional complexity in the enforcement of inter-patch continuity and, consequently, in the construction of efficient parallel solvers.

2. IETI-DP Solvers and Bidirectional Extraction Across Non-Matching Interfaces

The isogeometric tearing and interconnecting dual-primal (IETI-DP) framework is a domain decomposition paradigm for IgA, where the global domain is divided into non-overlapping patches. Each patch may have independent discretizations, and coupling across interfaces is governed by partitioning the degrees of freedom (DoFs) into interior, interface, and primal subsets. IETI-DP solvers assemble a global saddle-point system, typically solved with a preconditioned conjugate gradient method.

In non-matching configurations with T-junctions, standard DoF matching along edges is infeasible. Bidirectional T-patch extraction refers to formulating this coupling so that information is extracted in both directions across a T-junction, facilitating robust "glueing" between mismatched patches. This is achieved by introducing artificial interfaces, Lagrange multiplier constraints, and careful selection of primal (coarse) DoFs, ensuring that continuity is enforced even when the geometric overlap is partial or irregular (Schneckenleitner et al., 2021, Schneckenleitner et al., 2021).

3. Primal Degrees of Freedom and the “Fat Vertex” Concept

At T-junctions, the multiplicity of basis functions with non-zero support at a shared point increases, particularly as a function of the spline degree $p$. Standard practice in conforming patch decompositions assigns one primal DoF per vertex. For T-junctions, the number of DoFs associated with such a vertex—termed a "fat vertex"—corresponds to the totality of basis functions (across all adjacent patches) that do not vanish at that point.

The bidirectional extraction strategy includes all these basis functions at the T-junction vertex into the primal space. This over-enrichment ensures that, for any vertex (including T-junctions), all relevant coefficients are strongly coupled, thereby enforcing the required continuity conditions. Formally, if $u^{(k)}(x)$ denotes the basis function value at vertex $x$ for patch $k$, and $u^{(e,k)}(x)$ for its neighbor, continuity is imposed as $u^{(k)}(x) = u^{(e,k)}(x)$ for all $k, e$ sharing $x$. This fat vertex strategy guarantees robust coupling and stability, even as the number of non-vanishing basis functions grows linearly with $p$ (Schneckenleitner et al., 2021, Schneckenleitner et al., 2021).

4. Discontinuous Galerkin Coupling and Robustness

For T-junctions and other non-matching interfaces, symmetric interior penalty discontinuous Galerkin (SIPG) formulations are employed to couple patches. The SIPG method penalizes jumps in the solution and fluxes across interfaces, therefore weakly imposing continuity via interface terms:

$$m^{(k)}(u, v) = \int_{T(k,e)} \left( \delta \frac{a_k}{\min(h_k, h_e)} (u^{(e)} - u^{(k)}) (v^{(e)} - v^{(k)}) \right) ds$$

Here, $T(k,e)$ denotes an interface between patch $k$ and its neighbor $e$; $\delta$ is a penalty parameter; $a_k$ is a local coefficient; $h_k$, $h_e$ are mesh sizes. The SIPG formulation is agnostic to the precise geometric alignment, reinforcing the applicability of bidirectional T-patch extraction in non-matching settings (Schneckenleitner et al., 2021).

5. Theoretical and Numerical Performance Guarantees

A principal theoretical result underpinning bidirectional T-patch extraction is the preservation of favorable condition number bounds when using fat vertex primal DoFs. Specifically, for the preconditioned Schur complement system arising in IETI-DP, the bound is shown to be

$$\kappa(M_{SD} \cdot \hat{F}) \leq C_p (1 + \log p + \max_k \log(H_k/h_k))$$

where $H_k$ is the patch diameter, $h_k$ the local mesh size, $p$ the spline degree, and $C_p$ a $p$-independent constant. This matches the best-known bounds for the conforming case, demonstrating that neither the presence of T-junctions nor the bidirectional extraction strategy adversely affects convergence or robustness (Schneckenleitner et al., 2021, Schneckenleitner et al., 2021).

Numerical experiments confirm that the inclusion of extra (fat vertex) primal constraints leads to only a mild increase in the coarse space size but maintains or improves iteration counts and condition numbers compared to previous approaches. This robustness holds across varying spline degrees and mesh refinements.

6. Applications and Practical Implications

Bidirectional T-patch extraction is directly applicable to geometric domains with sliding interfaces, as commonly arises in electromechanical systems (e.g., simulations involving the rotor-stator interface in electric motors) (Schneckenleitner et al., 2021). In such scenarios, patches slide past each other, dynamically generating T-junctions; the described solvers efficiently accommodate the resulting non-matching interfaces and maintain numerical stability.

More generally, any IgA application involving complex domain decompositions, dynamic remeshing, or local geometric singularities benefits from the flexibility and robustness of bidirectional T-patch extraction. The strategy is also applicable in contexts where patches need to be coupled dynamically during simulation, further emphasizing its versatility.

7. Comparative Perspective and Future Directions

Relative to previous domain decomposition techniques requiring fully matching interfaces, bidirectional T-patch extraction with fat vertices offers enhanced flexibility, allowing arbitrary splitting and gluing of patches at T-junctions without deterioration in solver performance. While the coarse problem increases in size due to over-enrichment at fat vertices, the computational overhead is compensated by the improved conditioning and applicability to a much wider class of domains.

Potential future research directions include exploring more compact primal spaces that preserve robustness, optimized penalty parameter selection in SIPG coupling, and extensions to time-dependent or non-linear PDEs in IgA frameworks featuring evolving non-matching interfaces.

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In summary, bidirectional T-patch extraction—grounded in IETI-DP solvers, SIPG coupling, and the fat vertex concept—enables the accurate and robust resolution of PDEs on highly complex geometric domains, removing previous restrictions on patch alignment and supporting advanced engineering simulations with intricate, dynamic patch arrangements (Schneckenleitner et al., 2021, Schneckenleitner et al., 2021).