pyCAFE: A Finite Element Framework for Solving Acoustic Problems in Python

Daniele Fabbri; Fabio Bruzzone; Carlo Rosso

doi:10.5334/jors.677

pyCAFE: A Finite Element Framework for Solving Acoustic Problems in Python

Journal of Open Research Software

Volume 14 (2026): Issue 1

By: Daniele Fabbri , Fabio Bruzzone and Carlo Rosso

Open Access

|Jun 2026

Figures & Tables

Overall architecture of the pyCAFE framework and main software modules.

Table 1

Eigenfrequency comparison between pyCAFE and the analytical solution for the rectangular rigid-wall cavity. Results are reported for CQUAD4 and CQUAD8 elements using the balanced mesh size $h = 0.070 m$ .

MODE (m, n)	f_an [Hz]	f_CQUAD4 [Hz]	$ε 4$ [%]	f_CQUAD8 [Hz]	$ε 8$ [%]
(1,0)	171.5000	171.8601	0.2099	171.5003	0.0002
(0,1)	343.0000	345.8854	0.8412	343.0096	0.0028
(2,0)	343.0000	345.8854	0.8412	343.0096	0.0028
(1,1)	383.4857	386.2287	0.7153	383.4944	0.0023
(2,1)	485.0753	489.1558	0.8412	485.0890	0.0028
(3,0)	514.5000	524.2622	1.8974	514.5717	0.0139
(3,1)	618.3520	628.0825	1.5736	618.4180	0.0107
(0,2)	686.0000	709.1965	3.3814	686.2968	0.0433
(4,0)	686.0000	709.1965	3.3814	686.2968	0.0433
(1,2)	707.1126	729.7229	3.1976	707.4009	0.0408
Mean error		1.6880%		0.0163%

Histogram comparing the first 10 eigenfrequencies of the rectangular rigid-wall cavity obtained from the analytical solution, pyCAFE CQUAD4, and pyCAFE CQUAD8. The lower panel reports the relative error with respect to the analytical solution for the balanced mesh size $h = 0.070 m$ .

Table 2

$h$ -refinement convergence study for the rectangular rigid-wall cavity. The table reports the number of nodes, the mean relative eigenfrequency error over the first 10 modes, and the corresponding solver time for CQUAD4 and CQUAD8 elements.

h [m]	CQUAD4			CQUAD8
h [m]	NODES	MEAN ERR. [%]	TIME [s]	NODES	MEAN ERR. [%]	TIME [s]
0.200	18	10.2996	0.00	45	2.4886	0.01
0.140	40	5.8877	0.00	107	0.1851	0.01
0.100	66	3.3136	0.01	181	0.0606	0.02
0.070	120	1.6880	0.01	337	0.0163	0.04
0.050	231	0.8250	0.02	661	0.0040	0.09
0.035	450	0.4066	0.05	1305	0.0010	0.20

$h$ -refinement convergence of the mean eigenfrequency error over the first 10 modes. The dashed lines indicate the theoretical convergence rates $O (h^{2})$ for CQUAD4 and $O (h^{4})$ for CQUAD8. The solver time associated with each mesh is annotated next to the corresponding marker.

	pyCAFE	FEniCSx
Element type	CQUAD8 serendipity ( $p = 2$ )	Lagrange Q2 tensor-product
Mesh	Gmsh structured CQUAD8	`create_rectangle` $14 \times 7$
DOFs, modal	337	435
Hard-wall BC	Natural Neumann	Natural Neumann

Table 3

Comparison of the eigenfrequencies obtained with pyCAFE CQUAD8, FEniCSx Q2, and the analytical solution.

MODE (m, n)	f_an [Hz]	f_FEniCSx [Hz]	$ε F$ [%]	f_pyCAFE [Hz]	$ε P$ [%]
(1,0)	171.5000	171.5003	0.0002	171.5003	0.0002
(0,1)	343.0000	343.0096	0.0028	343.0096	0.0028
(2,0)	343.0000	343.0096	0.0028	343.0096	0.0028
(1,1)	383.4857	383.4943	0.0023	383.4944	0.0023
(2,1)	485.0753	485.0888	0.0028	485.0890	0.0028
(3,0)	514.5000	514.5717	0.0139	514.5717	0.0139
(3,1)	618.3520	618.4170	0.0105	618.4180	0.0107
(0,2)	686.0000	686.2968	0.0433	686.2968	0.0433
(4,0)	686.0000	686.2968	0.0433	686.2968	0.0433
(1,2)	707.1126	707.4007	0.0407	707.4009	0.0408
Mean error		0.0163%		0.0163%

Left: Grouped bar chart of the first 10 eigenfrequencies obtained from the analytical solution, FEniCSx Q2, and pyCAFE CQUAD8. Right: Modal comparison summary normalised by the FEniCSx values. Ratios below one indicate smaller values for pyCAFE.

Mode-shape comparison between pyCAFE CQUAD8 and FEniCSx Q2 for modes 1, 4, 6 and 7.

Left: Grouped bar chart of the first 10 eigenfrequencies for the zero pressure cavity, comparing the analytical solution, FEniCSx Q2, and pyCAFE CQUAD8. Right: Modal comparison metrics normalised by the FEniCSx values.

Mode-shape comparison between pyCAFE CQUAD8 and FEniCSx Q2 for modes 1, 2, 4 and 6 of the pressure-release cavity, with $p = 0$ on all four walls. The comparison confirms that the two solvers reproduce the same spatial pressure patterns.

Table 4

Comparison of the eigenfrequencies obtained with pyCAFE CQUAD8, FEniCSx Q2, and the analytical solution for the pressure-release cavity with $p = 0$ on all walls, using $h = 0.070 m$ .

MODE (m, n)	f_an [Hz]	f_FEniCSx [Hz]	$ε F$ [%]	f_pyCAFE [Hz]	$ε P$ [%]
(1,1)	383.4857	383.4943	0.0023	383.4944	0.0023
(2,1)	485.0753	485.0888	0.0028	485.0890	0.0028
(3,1)	618.3520	618.4170	0.0105	618.4180	0.0107
(1,2)	707.1126	707.4007	0.0407	707.4008	0.0408
(2,2)	766.9713	767.2411	0.0352	767.2438	0.0355
(4,1)	766.9713	767.2411	0.0352	767.2438	0.0355
(3,2)	857.5000	857.7805	0.0327	857.7932	0.0342
(5,1)	923.5558	924.3822	0.0895	924.3879	0.0901
(4,2)	970.1505	970.5703	0.0433	970.6082	0.0472
(1,3)	1043.1938	1045.3131	0.2032	1045.3136	0.2032
Mean error		0.0495%		0.0502%

Table 5

Eigenfrequencies, hard-wall cavity (Neumann walls), COMSOL versus pyCAFE CQUAD8 versus analytical ( $h = 0.070 m$ ).

MODE (m, n)	f_an [Hz]	f_COMSOL [Hz]	$ε C$ [%]	f_pyCAFE [Hz]	$ε P$ [%]
(1,0)	171.5000	171.6020	0.0595	171.5003	0.0002
(0,1)	343.0000	343.2108	0.0615	343.0096	0.0028
(2,0)	343.0000	343.2131	0.0621	343.0096	0.0028
(1,1)	383.4857	383.7219	0.0616	383.4944	0.0023
(2,1)	485.0753	485.3750	0.0618	485.0890	0.0028
(3,0)	514.5000	514.8599	0.0699	514.5717	0.0139
(3,1)	618.3520	618.7697	0.0675	618.4180	0.0107
(0,2)	686.0000	686.6336	0.0924	686.2968	0.0433
(4,0)	686.0000	686.7041	0.1026	686.2968	0.0433
(1,2)	707.1126	707.8204	0.1001	707.4008	0.0408
Mean error		0.0739%		0.0163%

Table 6

Eigenfrequencies, zero pressure on all the four sides of the cavity ( $p = 0$ on all walls), COMSOL versus pyCAFE CQUAD8 versus analytical ( $h = 0.070 m$ ).

MODE (m, n)	f_an [Hz]	f_COMSOL [Hz]	$ε C$ [%]	f_pyCAFE [Hz]	$ε P$ [%]
(1,1)	383.4857	383.7219	0.0616	383.4944	0.0023
(2,1)	485.0753	485.3750	0.0618	485.0890	0.0028
(3,1)	618.3520	618.7697	0.0675	618.4180	0.0107
(1,2)	707.1126	707.8204	0.1001	707.4008	0.0408
(2,2)	766.9713	767.6333	0.0863	767.2438	0.0355
(4,1)	766.9713	767.6953	0.0944	767.2438	0.0355
(3,2)	857.5000	858.2792	0.0909	857.7932	0.0342
(5,1)	923.5558	924.7370	0.1279	924.3879	0.0901
(4,2)	970.1505	971.0964	0.0975	970.6082	0.0472
(1,3)	1043.1938	1045.9333	0.2626	1045.3136	0.2032
Mean error		0.1051%		0.0502%

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DOI: https://doi.org/10.5334/jors.677 | Journal eISSN: 2049-9647

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Language: English

Page range: 48 - 48

Submitted on: Jan 13, 2026

Accepted on: May 28, 2026

Published on: Jun 22, 2026

Published by: Ubiquity Press

In partnership with: Paradigm Publishing Services

Keywords:

acoustics,

Finite Element Method,

Python,

Helmholtz equation,

frequency domain,

modal analysis,

CQUAD8,

computational acoustics,

open source

© 2026 Daniele Fabbri, Fabio Bruzzone, Carlo Rosso, published by Ubiquity Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 14 (2026): Issue 1

pyCAFE: A Finite Element Framework for Solving Acoustic Problems in Python

Figures & Tables

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Table 1

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Figure 3

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Table 4

Table 5

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Figure 8

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