Classical finite element methods (FEM) approximate curved boundaries by piecewise polynomials, which introduces geometric errors that degrade the accuracy of wave propagation simulations. Isogeometric analysis (IGA) overcomes this limitation by using the same B-spline basis functions for both exact geometry representation and solution approximation. This paper presents a complete Python implementation of IGA for solving the wave equation on curved domains, extending previous work from rectangular to curved geometries. The methodology includes three main steps. First, we construct exact B-spline parameterizations of a semicircular membrane and a quarter-annular plate, ensuring no geometric approximation error. Second, we discretize the weak Galerkin formulation in space using B-splines and in time using a fourth-order Runge-Kutta scheme. Third, we derive a generalized Courant-Friedrichs-Lewy (CFL) condition that incorporates the effective element size to account for the non-uniform Jacobian of curved mappings. Numerical experiments demonstrate that the method achieves optimal convergence rates. For quadratic B-splines, the observed rate reaches 2.98, very close to the theoretical value of 3. On the quarter-annulus test case, the IGA solution matches the analytical frequency with only 0.3% error, whereas standard FEM with a comparable number of degrees of freedom yields 2.1% error-a sevenfold improvement. The numerical CFL limit is accurately predicted by the generalized condition. The complete open-source Python code is provided in the appendix, enabling full reproducibility. This work lays a foundation for accurate wave propagation simulations on curved geometries in acoustics, elastodynamics, and seismology.
| Published in | Applied and Computational Mathematics (Volume 15, Issue 3) |
| DOI | 10.11648/j.acm.20261503.14 |
| Page(s) | 95-110 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Isogeometric Analysis, Wave Equation, B-splines, Curved Domains, Python, CFL Condition
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APA Style
Bah, T. M., Aguemon, U., Kalivogui, S., Diakhaby, A. (2026). Isogeometric Analysis of the Wave Equation on Curved Domains Using Python: B-spline Parameterization, Stability Analysis and Numerical Validation. Applied and Computational Mathematics, 15(3), 95-110. https://doi.org/10.11648/j.acm.20261503.14
ACS Style
Bah, T. M.; Aguemon, U.; Kalivogui, S.; Diakhaby, A. Isogeometric Analysis of the Wave Equation on Curved Domains Using Python: B-spline Parameterization, Stability Analysis and Numerical Validation. Appl. Comput. Math. 2026, 15(3), 95-110. doi: 10.11648/j.acm.20261503.14
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Download@article{10.11648/j.acm.20261503.14,
author = {Thierno Mamadou Bah and Uriel Aguemon and Siba Kalivogui and Aboubakary Diakhaby},
title = {Isogeometric Analysis of the Wave Equation on Curved Domains Using Python: B-spline Parameterization, Stability Analysis and Numerical Validation
},
journal = {Applied and Computational Mathematics},
volume = {15},
number = {3},
pages = {95-110},
doi = {10.11648/j.acm.20261503.14},
url = {https://doi.org/10.11648/j.acm.20261503.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261503.14},
abstract = {Classical finite element methods (FEM) approximate curved boundaries by piecewise polynomials, which introduces geometric errors that degrade the accuracy of wave propagation simulations. Isogeometric analysis (IGA) overcomes this limitation by using the same B-spline basis functions for both exact geometry representation and solution approximation. This paper presents a complete Python implementation of IGA for solving the wave equation on curved domains, extending previous work from rectangular to curved geometries. The methodology includes three main steps. First, we construct exact B-spline parameterizations of a semicircular membrane and a quarter-annular plate, ensuring no geometric approximation error. Second, we discretize the weak Galerkin formulation in space using B-splines and in time using a fourth-order Runge-Kutta scheme. Third, we derive a generalized Courant-Friedrichs-Lewy (CFL) condition that incorporates the effective element size to account for the non-uniform Jacobian of curved mappings. Numerical experiments demonstrate that the method achieves optimal convergence rates. For quadratic B-splines, the observed rate reaches 2.98, very close to the theoretical value of 3. On the quarter-annulus test case, the IGA solution matches the analytical frequency with only 0.3% error, whereas standard FEM with a comparable number of degrees of freedom yields 2.1% error-a sevenfold improvement. The numerical CFL limit is accurately predicted by the generalized condition. The complete open-source Python code is provided in the appendix, enabling full reproducibility. This work lays a foundation for accurate wave propagation simulations on curved geometries in acoustics, elastodynamics, and seismology.
},
year = {2026}
}
TY - JOUR T1 - Isogeometric Analysis of the Wave Equation on Curved Domains Using Python: B-spline Parameterization, Stability Analysis and Numerical Validation AU - Thierno Mamadou Bah AU - Uriel Aguemon AU - Siba Kalivogui AU - Aboubakary Diakhaby Y1 - 2026/06/10 PY - 2026 N1 - https://doi.org/10.11648/j.acm.20261503.14 DO - 10.11648/j.acm.20261503.14 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 95 EP - 110 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20261503.14 AB - Classical finite element methods (FEM) approximate curved boundaries by piecewise polynomials, which introduces geometric errors that degrade the accuracy of wave propagation simulations. Isogeometric analysis (IGA) overcomes this limitation by using the same B-spline basis functions for both exact geometry representation and solution approximation. This paper presents a complete Python implementation of IGA for solving the wave equation on curved domains, extending previous work from rectangular to curved geometries. The methodology includes three main steps. First, we construct exact B-spline parameterizations of a semicircular membrane and a quarter-annular plate, ensuring no geometric approximation error. Second, we discretize the weak Galerkin formulation in space using B-splines and in time using a fourth-order Runge-Kutta scheme. Third, we derive a generalized Courant-Friedrichs-Lewy (CFL) condition that incorporates the effective element size to account for the non-uniform Jacobian of curved mappings. Numerical experiments demonstrate that the method achieves optimal convergence rates. For quadratic B-splines, the observed rate reaches 2.98, very close to the theoretical value of 3. On the quarter-annulus test case, the IGA solution matches the analytical frequency with only 0.3% error, whereas standard FEM with a comparable number of degrees of freedom yields 2.1% error-a sevenfold improvement. The numerical CFL limit is accurately predicted by the generalized condition. The complete open-source Python code is provided in the appendix, enabling full reproducibility. This work lays a foundation for accurate wave propagation simulations on curved geometries in acoustics, elastodynamics, and seismology. VL - 15 IS - 3 ER -
Department of Mathematics, University of Kindia, Kindia, Guinea
Department of Mathematics, University of Kindia, Kindia, Guinea
West African Institute of Mathematics (IOAM), Abdel Nasser University of Conakry (UGANC), Conakry, Guinea
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