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Multi-resolution Isogeometric Analysis -- Efficient adaptivity utilizing the multi-patch structure

Stefan Tyoler, Stefan Takacs

TL;DR

This paper develops a practical adaptive strategy for multi-patch IgA that modestly increases the patch count while allowing patchwise grid sizes to preserve tensor-product structure and reuse existing code. It constructs a conforming global space through edge- and vertex-based trace constraints and provides a matrix-driven basis construction alongside rigorous a priori error estimates for both high and low regularity solutions. The approach is validated with numerical experiments (e.g., L-shaped domains and an electric motor) showing that adaptivity recovers near-optimal convergence rates and concentrates refinement where the solution exhibits reduced regularity or material discontinuities. Overall, the method offers a scalable, efficient pathway to high-order adaptive IgA on complex, CAD-based geometries with manageable increases in degrees of freedom and without global remeshing.

Abstract

Isogeometric Analysis (IgA) is a spline based approach to the numerical solution of partial differential equations. There are two major issues that IgA was designed to address. The first issue is the exact representation of domains stemming from Computer Aided Design (CAD) software. In practice, this can be realized only with multi-patch IgA, often in combination with trimming or similar techniques. The second issue is the realization of high-order discretizations (by increasing the spline degree) with numbers of degrees of freedom comparable to low-order methods. High-order methods can deliver their full potential only if the solution to be approximated is sufficiently smooth; otherwise, adaptive methods are required. In the last decades, a zoo of local refinement strategies for splines has been developed. The authors think that many of these approaches are a burden to implement efficiently and impede the utilization of recent advances that rely on tensor-product splines, e.g., concerning matrix assembly and preconditioning. The implementation seems to be particularly cumbersome in the context of multi-patch IgA. Our approach is to moderately increase the number of patches and to utilize different grid sizes on different patches. This allows reusing the existing code bases, recovers the convergence rates of other adaptive approaches and increases the number of degrees of freedom only marginally.

Multi-resolution Isogeometric Analysis -- Efficient adaptivity utilizing the multi-patch structure

TL;DR

This paper develops a practical adaptive strategy for multi-patch IgA that modestly increases the patch count while allowing patchwise grid sizes to preserve tensor-product structure and reuse existing code. It constructs a conforming global space through edge- and vertex-based trace constraints and provides a matrix-driven basis construction alongside rigorous a priori error estimates for both high and low regularity solutions. The approach is validated with numerical experiments (e.g., L-shaped domains and an electric motor) showing that adaptivity recovers near-optimal convergence rates and concentrates refinement where the solution exhibits reduced regularity or material discontinuities. Overall, the method offers a scalable, efficient pathway to high-order adaptive IgA on complex, CAD-based geometries with manageable increases in degrees of freedom and without global remeshing.

Abstract

Isogeometric Analysis (IgA) is a spline based approach to the numerical solution of partial differential equations. There are two major issues that IgA was designed to address. The first issue is the exact representation of domains stemming from Computer Aided Design (CAD) software. In practice, this can be realized only with multi-patch IgA, often in combination with trimming or similar techniques. The second issue is the realization of high-order discretizations (by increasing the spline degree) with numbers of degrees of freedom comparable to low-order methods. High-order methods can deliver their full potential only if the solution to be approximated is sufficiently smooth; otherwise, adaptive methods are required. In the last decades, a zoo of local refinement strategies for splines has been developed. The authors think that many of these approaches are a burden to implement efficiently and impede the utilization of recent advances that rely on tensor-product splines, e.g., concerning matrix assembly and preconditioning. The implementation seems to be particularly cumbersome in the context of multi-patch IgA. Our approach is to moderately increase the number of patches and to utilize different grid sizes on different patches. This allows reusing the existing code bases, recovers the convergence rates of other adaptive approaches and increases the number of degrees of freedom only marginally.
Paper Structure (15 sections, 25 theorems, 99 equations, 14 figures)

This paper contains 15 sections, 25 theorems, 99 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Isogeometric Galerkin method
  3. Adaptivity and non-conforming multi patch discretization
  4. Construction of a basis
  5. Approximation error estimates
  6. Geometric setup
  7. Patch-local quasi-interpolation
  8. Traces for edges and vertices
  9. Extensions for vertices and edges
  10. Overall error estimate
  11. Overall error estimate for the low-regularity case
  12. Numerical experiments
  13. L-shape domain
  14. Electric Motor
  15. Conclusions

Key Result

Lemma 3.1

Provided that the geometry representation satisfies ass:map and the patch $\Omega_k$ is replaced by the patches $\tilde{\Omega}_{k,i}:=\tilde{G}_{k,i}(\widehat{\Omega})$ with $\tilde{G}_{k,i}$ as in eq:patchsplit for $i\in\{1,2,3,4\}$, then the new geometry representation satisfies ass:map with the

Figures (14)

  • Figure 1: Local refinement of patch $\Omega_2$.
  • Figure 2: Schematic representation of constraints
  • Figure 3: Example with $5$ patches $\Omega_k$, $7$ edges $\Gamma_i$, T-junction $\mathbf{x}_1$ and corner vertex $\mathbf{x}_2$.
  • Figure 4: Extensions for the T-junction $\mathbf{x}_1$ are defined on $\Omega_2$ (support in green) and $\Omega_3$ (support in blue); their depth into the finer patches is adjusted by $h_{\mathrm{k}(\mathbf{x}_m)}=h_1$, still aligned with the grids on which they are defined. The extensions agree on the adjacent edge $\Gamma_3$ and vanish on all other vertices.
  • Figure 5: The edge extensions (support in green) extend into the patch with the finer grid, here $\Omega_2$; their depth is adjusted to the grid size of the patch with the coarser grid, here $\Omega_1$, still aligned with the grid on which it is defined. The extension vanishes on all corners and on all other edges.
  • ...and 9 more figures

Theorems & Definitions (50)

  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Proof
  • Lemma 4.2
  • Proof
  • Lemma 4.3
  • Proof
  • Lemma 4.4
  • Proof
  • ...and 40 more