igaops

Matrix-free weighted quadrature operators for Isogeometric Analysis

igaops provides reusable numerical building blocks for Isogeometric Analysis (IGA) based on weighted quadrature (WQ) and matrix-free (MF) techniques. The algorithms are largely based on developments carried out during the author's PhD thesis and on methods described in the published literature (see Bibliography).

The library is written in pure Python and relies only on numpy, scipy, and geomdl. It is designed to be solver- and application-agnostic: it exposes numerical operators rather than a complete simulation pipeline. Boundary conditions, material models, and problem-specific solvers are intentionally out of scope.

What this package provides

• Weighted quadrature rules — computation of weights and points from univariate knot vectors.

• L-mode products — efficient tensor-product contractions between multi-dimensional arrays (sum-factorisation / L-mode matrix–vector products).

• Matrix-free operators — stiffness and mass matrix–vector products performed without explicitly assembling the global matrix, exploiting the tensor-product structure of IGA basis functions.

• Matrix assembly — classical sparse assembly for cases where an explicit matrix is needed, for example for direct solvers or debugging.

• Preconditioners — construction and application of the fast diagonalisation preconditioner for IGA, based on the Sylvester equation approach of Sangalli & Tani.

• Geometry wrapper — a thin class around geomdl objects that exposes matrix-free evaluation of the Jacobian, Hessian, and related geometric quantities needed by the operators above.

Scope

igaops currently focuses on tensor-product B-spline and NURBS discretizations and provides low-level numerical tools that can be used to build IGA solvers and applications.

It does not aim to provide a complete finite element or simulation framework.

What this package does not provide

Boundary condition handling, material libraries, load vectors, and time integration schemes are deliberately excluded. These components are application-specific and are best developed at a higher level, for example within YETI, the laboratory's Fortran/Python simulation framework.

Requirements

• Python ≥ 3.10
• numpy
• scipy
• geomdl

The required dependencies are installed automatically.

Optional dependencies for post-processing and visualisation:

• matplotlib
• pandas
• seaborn

It is also recommended to install ParaView for .vtk output visualisation.

Installation

Install the latest released version from PyPI:

pip install igaops

To include the optional visualisation dependencies:

pip install "igaops[viz]"

For a development install from a local clone:

git clone https://github.com/JoaquinCORNEJO/igaops.git

cd igaops

pip install -e ".[viz]"

Development

This project uses Black for code formatting, mypy for static type checking, and pytest for testing.

For a local development environment:

pip install -e ".[dev]"

Format the source code with:

black src/

Run static type checking with:

mypy src/

Run the test suite with:

pytest

Citation

If you use igaops in academic work, please cite the relevant publications and/or thesis describing the methods implemented in the library.

License

igaops is distributed under the GNU Lesser General Public License v2.1 or later (LGPL-2.1-or-later).

See the LICENSE file for the complete license text.

Contact

Questions and feedback are welcome:

joaquin.cofu@gmail.com

Bibliography

The algorithms implemented in igaops are based on the following references.

B-Splines and NURBS

• L. Piegl — The NURBS Book

• J. Cottrell, T. J. R. Hughes, Y. Bazilevs — Isogeometric Analysis: Toward Integration of CAD and FEA

Weighted quadrature and matrix-free methods

• F. Calabrò, G. Sangalli, M. Tani — Fast formation of isogeometric Galerkin matrices by weighted quadrature

• G. Sangalli, M. Tani — Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

• R. Hiemstra et al. — Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity

Preconditioners and fast diagonalisation

• G. Sangalli, M. Tani — Isogeometric preconditioners based on fast solvers for the Sylvester equation

• M. Montardini — Preconditioners for isogeometric analysis (PhD thesis)

Tensor products and sum-factorisation

• T. Kolda, B. Bader — Tensor decompositions and applications

• P. Antolin et al. — Efficient matrix computation for tensor-product isogeometric analysis