On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis
Abstract
The efficacy of hierarchical B-spline bases in solving high-dimensional partial differential equations has been demonstrated in several studies. However, the process of assembling isogeometric Galerkin matrices remains computationally challenging with standard methods, particularly as the dimension and the spline degree of the basis increase. In this work, we demonstrate the possibility of exploiting the structure of the system matrices to derive a fast assembly method. First, we introduce the Hadamard format, a compact representation of the Galerkin matrices based on low-rank approximation of the integral kernels and univariate integration. Second, we present an efficient algorithm for evaluating the Hadamard format and constructing the full system matrix. The algorithm relies on a dedicated data structure, denoted retraction tree, for representing the index set of the active basis functions, and exploits the restricted deep-first search to quickly compute the sparsity pattern of the full matrix. The complexity analysis shows that the proposed method achieves quasi-optimal complexity with respect to the matrix size. Our experiments, conducted using the C++ library G+Smo, validate the derived complexity and indicate that the proposed method significantly enhances the assembly process.
Related
BibTeX
@article{giannelli2026hadamard,
title = {{On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis}},
author = {Carlotta Giannelli and Angelos Mantzaflaris and Mattia Matucci},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2026},
month = oct,
doi = {10.1016/j.cma.2026.119285},
url = {https://hal.science/hal-05805454v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis
AU - Carlotta Giannelli
AU - Angelos Mantzaflaris
AU - Mattia Matucci
PY - 2026
DA - 2026/10/26
JO - Computer Methods in Applied Mechanics and Engineering
DO - 10.1016/j.cma.2026.119285
UR - https://hal.science/hal-05805454v1
AB - The efficacy of hierarchical B-spline bases in solving high-dimensional partial differential equations has been demonstrated in several studies. However, the process of assembling isogeometric Galerkin matrices remains computationally challenging with standard methods, particularly as the dimension and the spline degree of the basis increase. In this work, we demonstrate the possibility of exploiting the structure of the system matrices to derive a fast assembly method. First, we introduce the Hadamard format, a compact representation of the Galerkin matrices based on low-rank approximation of the integral kernels and univariate integration. Second, we present an efficient algorithm for evaluating the Hadamard format and constructing the full system matrix. The algorithm relies on a dedicated data structure, denoted retraction tree, for representing the index set of the active basis functions, and exploits the restricted deep-first search to quickly compute the sparsity pattern of the full matrix. The complexity analysis shows that the proposed method achieves quasi-optimal complexity with respect to the matrix size. Our experiments, conducted using the C++ library G+Smo, validate the derived complexity and indicate that the proposed method significantly enhances the assembly process.
SN - 0045-7825
SN - 1879-2138
ER -