Multi-Patch Isogeometric Analysis

• Multi-Patch Isogeometric Analysis is a discretization framework that decomposes a computational domain into independently parameterized patches to represent complex geometries accurately.
• It employs various interface coupling models—such as conforming, discontinuous Galerkin, and AS-G1 smooth coupling—to ensure the requisite continuity and optimal convergence across patches.
• Advanced solver architectures, adaptivity, and neural extensions enable efficient handling of higher-order PDEs and complex shell models while maintaining robust performance.

Searching arXiv for recent and foundational papers on multi-patch isogeometric analysis. Multi-patch isogeometric analysis is the spline- or NURBS-based discretization of a computational domain represented as a union of several geometric patches, each with its own parameterization, rather than by a single global map. In this setting, the same class of basis functions is used for geometry and analysis, but the patch decomposition turns geometric representation into an interface problem: neighboring traces may be matched strongly, coupled weakly by discontinuous Galerkin or Nitsche terms, or constrained to support globally $C^1$- or $C^2$-smooth spaces required by higher-order PDEs and shell models (Langer et al., 2014).

1. Patchwise geometry, parameterizations, and discrete spaces

A standard starting point is the decomposition

$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$

The admissible patch adjacencies are the usual ones: for $k\neq \ell$, the intersection of patch closures is empty or consists of a common vertex, a common edge, or, in three dimensions, a common face (Schneckenleitner et al., 2021). On each patch, tensor-product spline or NURBS spaces are defined on the parameter domain and pulled back to the physical patch, so the exact CAD geometry and the trial space share the same parametric structure (Langer et al., 2014).

Two global-space paradigms recur throughout the literature. In the fully matching conforming case, interface traces match exactly and continuity is imposed strongly across neighboring patches (Schneckenleitner et al., 2021). In discontinuous or multiplier-based formulations, the global trial space is first assembled as the product of patchwise spaces,

$V_h=\prod_{k=1}^N V_h^{(k)},$

so functions are generally discontinuous across interfaces unless continuity is imposed separately (Bosy et al., 2020). This distinction is not merely algebraic: it determines whether interface treatment is encoded in the basis itself or delegated to dG, Nitsche, tearing-and-interconnecting, or other coupling operators.

The same geometric framework is used for volumetric diffusion, linear elasticity, Stokes flow, wave propagation, Kirchhoff–Love shells, and surface PDEs on open and closed surfaces (Langer et al., 2014). What changes across these applications is the regularity demanded at patch interfaces and the solver technology needed to make the resulting systems practical.

2. Interface coupling models

The interface problem in multi-patch IgA is handled by several structurally distinct couplings (Takacs, 2019, Antolin et al., 2019, Huynh et al., 2023).

Coupling model	Interface requirement	Typical role
Conforming	Fully matching traces	Standard Galerkin coupling
SIPG / dG	No matching requirement	Non-conforming patches, T-junctions, wave problems
Nitsche on visible interfaces	Overlapping visible parts	Boolean union, trimming
AS-$G^1$ smooth coupling	Geometry compatible with global $C^1$ spaces	Biharmonic and Kirchhoff–Love formulations
Bending strip coupling	Standard multi-patch $C^0$ interfaces plus fictitious strip	Kirchhoff–Love shells on CAD-like geometries

If the discretization spaces agree on interfaces, conforming coupling is possible; otherwise non-conforming discretizations utilizing discontinuous Galerkin approaches are required (Takacs, 2019). In the SIPG setting, jump and average terms together with a penalty term enforce stability on nonmatching interfaces (Langer et al., 2014). For domains obtained by Boolean union of multiple independent, generally non-conforming and trimmed spline patches, visible parts are coupled by Nitsche’s method, and a minimal stabilization is added to address instability on interfaces shared by small trimmed elements (Antolin et al., 2019).

A common misconception is that matching nodal values alone suffices for an interface-compatible high-order method. In multi-patch C-IGA, nodal compatibility guarantees a unique global nodal unknown set, but it does not guarantee pointwise continuity across the interface; $G^0$ compatibility is the stronger condition needed to recover optimal PDE convergence, and without it the interface term can dominate and reduce the convergence order to first order (Zhang et al., 2024).

3. Globally smooth multi-patch spaces

Higher-order PDEs and Kirchhoff–Love shells require more than global $C^0$ continuity because the weak form involves second derivatives. On generic multi-patch geometries, naive $C^2$0 gluing can over-constrain the spline space; in the worst cases, the space becomes fully locked to linears at the patch interface (Collin et al., 2015). This is the motivation for analysis-suitable $C^2$1 geometry.

AS-$C^2$2 parameterizations are a structured subclass of $C^2$3 geometries in which the gluing data are restricted so that globally $C^2$4-smooth isogeometric spaces retain good approximation power. The two-patch planar theory shows that bilinear two-patch parameterizations are AS-$C^2$5, while non-AS geometries prevent optimal convergence of $C^2$6 spaces (Collin et al., 2015). For generic planar multi-patch domains, an AS-$C^2$7 parameterization can be constructed by a quadratic optimization problem with linear side constraints, preserving the same boundary, the same vertices, and the same first derivatives at the vertices while remaining as close as possible to the initial geometry; numerical tests confirm optimal convergence on AS-$C^2$8 domains and severely reduced convergence on generic parameterizations (Kapl et al., 2017).

On AS-$C^2$9 shell surfaces, the global $$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$0-smooth space

$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$1

can be realized through locally supported patch, edge, and vertex functions (Farahat et al., 2022). This construction extends to geometries with extraordinary vertices and to boundary vertices where some patch junctions are only $$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$2. The same entity-based pattern appears at one higher regularity: for planar bilinearly parameterized multi-patch domains, globally $$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$3-smooth spaces are built from patch, edge, and vertex functions and used for collocation of the Poisson equation (Kapl et al., 2019) and for the triharmonic equation (Kapl et al., 2018). In both cases, the basis is entirely local with small support, and the construction is described as simple and uniform across multi-patch configurations.

The major conceptual point is that smooth multi-patch analysis is not determined by geometry alone. It depends on whether the parameterization class supports traces and transversal derivatives with the right spline structure across interfaces; AS-$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$4 is precisely the condition introduced to guarantee that property.

4. Kirchhoff–Love shells on multi-patch domains

Multi-patch shell analysis is a particularly stringent test of interface regularity because the Green–Lagrange strains split into membrane and bending parts,

$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$5

so curvature terms bring second derivatives into the weak form (Farahat et al., 2022, Huynh et al., 2023). Two main strategies appear in the literature.

One strategy is exact global $$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$6 coupling on AS-$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$7 mid-surfaces. The shell discretization is then based simultaneously on an AS-$$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$8 multi-patch representation of the mid-surface and on a globally exactly $$\overline{\Omega}=\bigcup_{k=1}^K \overline{\Omega^{(k)}}, \qquad \Omega^{(k)}=G_k(\widehat\Omega), \qquad \widehat\Omega=(0,1)^d.$$9-smooth multi-patch spline space. This yields a parameter-free, strong-coupling IGA method for linear and nonlinear Kirchhoff–Love shells, including geometries with extraordinary vertices and boundary vertices where some patches meet only $k\neq \ell$0 (Farahat et al., 2022). The method gives smooth stress fields and handles geometrically complex multi-patch structures, but the chosen subspace $k\neq \ell$1 is smaller than the full $k\neq \ell$2 space, so convergence is typically slower than single-patch or penalty-coupled solutions.

A second strategy avoids strict global $k\neq \ell$3 construction and instead makes a $k\neq \ell$4 multi-patch shell practical by the bending strip method. In this formulation, adjacent patches are connected by a fictitious strip over the interface, and the total virtual work is modified to

$k\neq \ell$5

The strip is quadratic in the direction normal to the interface and linear along the interface, and its constitutive matrix is chosen so that the strip transmits bending moments across the interface and suppresses the artificial hinge that would otherwise appear at a $k\neq \ell$6 patch boundary (Huynh et al., 2023). This removes the need for strict global $k\neq \ell$7 continuity between patches and allows standard multi-patch NURBS CAD geometries to be used directly in a Kirchhoff–Love setting.

The same shell framework embeds arbitrary three-dimensional nonlinear material laws by a plane-stress reduction at each Gauss point through a nested Newton iteration, with the thickness stretch $k\neq \ell$8 as the unknown. Hyperelasticity and finite-strain plasticity are both supported, and time integration is carried out by a backward-Euler return-mapping algorithm (Huynh et al., 2023). For implementation, Bézier extraction,

$k\neq \ell$9

restores a finite-element-like local structure while preserving exact NURBS geometry and the required intra-patch smoothness. This combination of multi-patch coupling, local assembly, and general constitutive updates is the main implementation-oriented contribution of the formulation.

5. Solver architectures and computational scalability

The linear systems generated by multi-patch IgA are commonly attacked by domain decomposition, multigrid, or tensor-structured solvers. For conforming three-dimensional multi-patch discretizations, IETI-DP methods treat each patch as a subdomain and solve the interface Schur complement with PCG and a scaled Dirichlet preconditioner. Numerical evidence indicates in 3D the same favorable behavior known in 2D, but the choice of primal constraints is decisive: vertex values alone are the worst choice, while edge averages are consistently the best or among the best (Schneckenleitner et al., 2021).

Inexact local solvers that exploit tensor-product structure appear in the all-floating AF-IETI formulation. There, even Dirichlet conditions are imposed by multipliers so that the patch problems preserve a Sylvester-like tensor form suitable for Fast Diagonalization. The resulting preconditioned MINRES system satisfies a standard FETI-type condition number bound

$V_h=\prod_{k=1}^N V_h^{(k)},$0

with $V_h=\prod_{k=1}^N V_h^{(k)},$1 independent of $V_h=\prod_{k=1}^N V_h^{(k)},$2 and $V_h=\prod_{k=1}^N V_h^{(k)},$3, and numerical tests show favorable dependence on the spline degree $V_h=\prod_{k=1}^N V_h^{(k)},$4 (Bosy et al., 2020). In the non-conforming SIPG setting with T-junctions, the corresponding IETI-DP analysis introduces fat vertices, meaning that all basis functions non-zero at a T-junction are made primal; with that choice, the condition number bound coincides with the conforming-case bound (Schneckenleitner et al., 2021).

For saddle-point systems, the same patchwise viewpoint persists. In multi-patch Taylor–Hood discretizations of the Stokes problem, stability can be lifted from single patches to multi-patch domains, but the discrete inf-sup constant deteriorates with the global domain shape. The IETI-DP solver is designed specifically to avoid that global geometry-dependent effect: its condition-number estimate depends on local patch stability constants $V_h=\prod_{k=1}^N V_h^{(k)},$5, not on the global inf-sup constant $V_h=\prod_{k=1}^N V_h^{(k)},$6 (Sogn et al., 2022).

Multigrid provides a separate line of development. For conforming and SIPG non-conforming multi-patch IgA, the subspace corrected mass smoother yields robust performance in grid size and spline degree, and a hybrid smoother combining SCMS with Gauss–Seidel gives the best practical compromise (Takacs, 2019). The same patch/edge/vertex decomposition underlies a parallel MPI implementation of additive-Schwarz multigrid that remains robust with respect to both mesh refinement and spline degree and scales well on the Yeti footprint and Fichera corner benchmarks (Hofer et al., 2018).

For transient hyperbolic problems, multi-patch dG is especially attractive because the global mass matrix is block diagonal. On each patch, weight-adjusted inverse mass matrices restore a tensor-product structure on curved geometries while retaining high-order accuracy and semi-discrete energy stability, and spline spaces yield less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces (Chan et al., 2017). A different compression route is provided by low-rank tensor methods: for conforming multipatch linear elasticity, overlapping subdomains built from unions of neighboring patches enable Tucker-format local solvers inside an overlapping Schwarz preconditioner, and the reported memory usage on the finest grids is around $V_h=\prod_{k=1}^N V_h^{(k)},$7 of the full solution while iteration counts remain uniformly bounded with respect to both mesh size and spline degree (Montardini et al., 2023).

6. Adaptivity, reparameterization, and neural extensions

A persistent difficulty in multi-patch IgA is adaptivity without sacrificing tensor-product structure. One response is multi-resolution patch refinement: instead of introducing HB-, THB-, LR-, or T-spline technology, marked patches are split into sub-patches and assigned smaller local grid sizes. This moderately increases the number of patches, preserves tensor-product structure on each patch, reuses existing code bases, recovers the convergence rates of other adaptive approaches, and increases the number of degrees of freedom only marginally (Tyoler et al., 2024). The construction relies on nested trace spaces across interfaces, so the resulting discretization remains conforming even though the local resolutions differ.

A second response is $V_h=\prod_{k=1}^N V_h^{(k)},$8-adaptivity by reparameterization. Starting from bilinearly parameterized multi-patch domains, a trained residual neural network is applied to graph surfaces of the approximated solution and its derivatives to predict a biquadratic multi-patch parameterization that minimizes a best-approximation surrogate while preserving the tensor-product structure of the underlying spline space (Rios et al., 2024). The reported experiments show a vast improvement of the approximation error for different PDE problems on multi-patch domains.

Learning-based solvers also now use the multi-patch structure itself as the computational scaffold. In the multi-patch isogeometric neural solver, each NURBS patch carries its own neural network on the reference domain, conformity across interfaces is enforced by dedicated interface neural networks, Dirichlet boundary conditions are imposed strongly by a custom output layer, and training minimizes the discretized energy functional derived from the weak form (Tresckow et al., 29 Sep 2025). The method gives excellent agreement with reference solutions on a quadrupole magnet and a nonlinear contact-mechanics holder, but the paper is explicit that it is not yet computationally competitive with classical FEM or IGA, that it presently enforces only $V_h=\prod_{k=1}^N V_h^{(k)},$9 continuity, and that automatic patch-conforming CAD generation is still missing.

Across these developments, multi-patch IgA is not a single method but a family of geometry-aware discretization strategies. The patch decomposition can be the source of smoothness constraints, weak interface operators, local tensor structure, adaptive resolution, or neural domain decomposition; the central research problem is to exploit that structure without losing approximation power, stability, or solver efficiency.