The boundary element method (BEM) is a popular method for solving acoustic wave propagation problems, especially those in exterior domains, owing to its ease in handling radiation conditions at infinity. However, BEM models must meet the requirement of 6–10 elements per wavelength, using the conventional constant, linear, or quadratic elements. Therefore, a large storage size of memory and long solution time are often needed in solving higher-frequency problems. In this work, we propose two new types of enriched elements based on conventional constant boundary elements to improve the computational efficiency of the 2D acoustic BEM. The first one uses a plane wave expansion, which can be used to model scattering problems. The second one uses a special plane wave expansion, which can be used to model radiation problems. Five examples are investigated to show the advantages of the enriched elements. Compared with the conventional constant elements, the new enriched elements can deliver results with the same accuracy and in less computational time. This improvement in the computational efficiency is more evident at higher frequencies (with the nondimensional wave numbers exceeding 100). The paper concludes with the potential of our proposed enriched elements and plans for their further improvement.

The finite element method (FEM) [

Researchers have developed numerous techniques in the last few decades to improve the efficiency of the BEM solutions, including the fast multipole method for potential, elastostatic, and acoustic problems (see some works in [

Using enriched elements is one way to reduce the size of the BEM matrix while maintaining the accuracy of the model. One of the methods used to enrich elements is the partition-of-unity method (PUM). The PUM was proposed by Melenk et al. in 1996 [

In solving acoustic problems, however, the desired results of acoustic quantities are often at locations far away from the boundary of the domain. The use of higher-order elements may not be critical. Therefore, constant boundary elements are often applied in the acoustic BEM because they are easier to implement (regarding the treatment of corner problems and analytical integration of the singular integrals). They are often used with fast solution methods for the acoustic BEM owing to these advantages [

In this study, we propose two types of enrichment constant elements in the BEM for 2D acoustic wave problems: 1) a plane wave expansion with constant elements, for modeling scattering problems; and 2) a special plane wave expansion with constant elements, for solving radiation problems. Two verification problems are used to study the performance of the proposed elements regarding computational efficiency and accuracy. The numerical results show that enriched constant elements perform well in both scattering and radiation problems and that using even a single-term expansion can reduce errors markedly. The plane-wave-expansion elements are also used in solving two larger-scale problems and an acoustic meta-material model. The proposed work is novel in that it uses the enriched element concept with constant elements for the acoustic BEM. The enriched constant elements are easy to implement, more accurate, and more efficient compared with the regular constant elements.

The rest of this paper is organized as follows: In

Consider the propagation of time-harmonic acoustic waves in a homogeneous isotropic acoustic medium Ω (either finite or infinite), governed by the following Helmholtz equation (e.g., Chapter 6 in [

The boundary conditions for acoustic problems can be the Dirichlet boundary condition, Neumann boundary condition, or Robin boundary condition. For exterior domain problems, the Sommerfeld condition at infinity must also be satisfied.

Using Green’s second identity, and letting point

In the BEM, the boundary can be discretized using constant, linear, quadratic, or other higher-order elements. For solving acoustic problems, we use the constant elements to yield the following discretized form of the BIE (BEM equations) [

We collocate

To introduce the enrichment, we use a new function on a constant element to better represent the acoustic field on the element. For example, for scattering problems, we describe the sound pressure on the element as the sum of a series of plane waves as follows (instead of assuming it is constant on the element):
_{l} is the amplitude of the pressure at point

The integral on element

For radiation problems, it is found that the use of plane wave expansions as in

In

The unknowns on a traditional constant element, an enriched element for scattering problems, and an enriched element for radiation problems are shown in _{l} in the direction

Therefore, the linear system of the BEM equations with the enriched elements becomes

From

We consider a problem with a circular scatterer (an infinitely long cylinder) and another problem with a pulsating cylinder, both in a 2D infinite domain, to verify the correctness of the enriched elements presented in the previous section. In both cases, the circular domain has a radius of 1 m and is discretized with 60 elements initially. A field surface is defined by a square region of 15 m × 15 m around the cylinder and is placed with 2,080 field points to measure the error of the calculation. We use the

The sound-hard boundary condition is considered for the problem of scattering from the cylinder, with an incident plane wave propagating in the +

The results using the enriched elements with

The contour plots of the computed sound pressure distributions on the field surface using the analytical solution, conventional constant elements, and the enriched constant elements with

An enriched constant element with three expansion terms has the same number of DOFs as that of three conventional constant elements. Therefore, we next compare the results for the scattering cylinder using these two types of elements with the same number of DOFs.

The computing time for the two types of elements with the same number of DOFs is shown in

Conventional constant elements | Sixty enriched constant elements | ||||
---|---|---|---|---|---|

No. of elements | Time (s) | Time (s) | |||

60 | 36 | 130.27 | 1 | 78 | 129.06 |

180 | 38 | 193.70 | 3 | 66 | 149.18 |

300 | 38 | 249.24 | 5 | 76 | 154.55 |

420 | 78 | 326.48 | 7 | 76 | 175.56 |

The errors when using 420 conventional constant elements and 60 enriched constant elements with seven expansion terms (number of DOFs = 60 × 7 = 420) are shown in

Next, the model of a pulsating cylinder (with a radius of 1 m) is considered, with a constant pressure boundary condition (

We first test the case using the same type of plane-wave-expansion-enriched constant elements as used for the scattering problem. The errors of the results are shown in

Based on this observation, to more effectively model the radiation problems with the enriched elements, we turn to the special plane-wave-expansion-enriched constant elements, as introduced in

The contour plots of the computed sound pressure on the field surface at

In this section, to verify the effectiveness and efficiency of using the proposed enriched elements, we consider models of three problems on domains with more complicated geometries. To verify the accuracy of the enriched elements, we use the

First, we study the scattering problem from a regular array of 25 (5 × 5) infinitely long circular cylinders. To describe the geometry and field accurately in this case, we use a large number of plane-wave-enriched elements. This model is used to verify whether the enriched elements are suitable for solving larger-scale BEM models.

For scattering from a hard surface, the boundary condition is

The contour plots of the sound pressure on the field surface with different numbers of plane-wave-expansion terms are shown in

The comparison of the numbers of elements, total numbers of DOFs, percent differences in the solutions, and computing times when using the two methods is shown in

No. of elements on each cylinder | No. of expansion terms | No. of DOFs | Time (s) | |
---|---|---|---|---|

1,750 | – | 43,750 | – | 25,970.20 |

400 | 1 | 10,000 | 5.339% | 2,806.57 |

400 | 3 | 30,000 | 0.243% | 5,998.07 |

400 | 5 | 50,000 | 0.249% | 10,926.24 |

In this case, to calculate a radiation problem and investigate the effectiveness of the special plane-wave-enriched elements, we use the same model as in the previous section. For this purpose, the number of elements for each circle is set to 300 in the current calculations, which is sufficient for obtaining converged results using the conventional constant elements for the radiation problem when

The contour plots of sound pressure computed for the model with multiple cylinders at

In this case, we study the scattering problem of an acoustic meta-material model using the enriched elements to demonstrate the feasibility of the enriched elements in solving problems in domains with complex geometries. The rule for selecting angles of the expansion terms for the enriched elements in this case is related not only to the incidence angle of the plane wave but also to the direction normal to the boundary. We choose the first angle

The structure of the meta-material model is shown in

The contour plots of the acoustic wave propagation under the influence of structures in the meta-material model are shown in

No. of elements | No. of expansion terms | No. of DOFs | Time (s) | |
---|---|---|---|---|

2,928 | – | 2,928 | – | 739.85 |

366 | 1 | 366 | 9.148% | 101.99 |

366 | 3 | 1,098 | 1.678% | 134.86 |

366 | 5 | 1,830 | 0.386% | 162.67 |

In

No. of expansion terms | No. of elements | No. of DOFs | Time (s) | |
---|---|---|---|---|

1 | 2,562 | 2,562 | 0.727% | 721.86 |

3 | 732 | 2,196 | 0.520% | 266.23 |

5 | 366 | 1,830 | 0.386% | 162.67 |

Based on the results in

In this paper, two types of enriched constant elements to solve 2D acoustic wave problems are proposed: 1) a plane wave expansion, for scattering problems, and 2) a special plane wave expansion for radiation problems. These new enriched elements can be applied to solve acoustic wave problems at higher frequencies as demonstrated by the verification and application cases. To verify the accuracy of these enriched elements, two verification cases have been studied. To show their capability to solve problems with more complex geometries, three application cases have been studied. It has shown that using the plane-wave-expansion-enriched constant elements is more efficient than using the conventional constant elements for solving scattering and radiation problems with the same number of DOFs. It has also shown that the error in the results is fewer when using the enriched elements than when using the conventional constant elements at the same frequency when a boundary mesh is given. Our work is novel in using enriched elements with constant elements, making it easy to implement in the treatment of corners of geometries and analytical integration of singular integrals.

The plane-wave-expansion-enriched elements and the special-plane-wave-expansion-enriched elements still need to be improved regarding their implementations and solutions of the BEM system of equations. The fictitious eigenfrequency problems existing with the conventional BIE need to be addressed. These enriched elements can also be extended to solve 3D acoustic wave problems, although the implementation will be more complicated. The solution of the BEM equations with the enriched boundary elements can be accelerated using the fast multipole method [

The authors would like to thank the anonymous reviewers and journal editors for providing useful comments and suggestions.

The authors would like to thank the following for their financial support for this work: the National Natural Science Foundation of China (

The authors confirm contribution to the paper as follows: study conception and design: Zonglin Li, Yijun Liu; program and data collection: Zonglin Li; analysis and interpretation of results: Zonglin Li, Zhenyu Gao, Yijun Liu; draft manuscript preparation: Zonglin Li, Yijun Liu. All authors reviewed the results and approved the final version of the manuscript.

None.

The authors declare that they have no conflicts of interest to report regarding the present study.