# On the Robustness and Efficiency of the Plane-Wave-Enriched FEM with Variable q-Approach on the 2D Room Acoustics Problem

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Study Purpose

## 2. Theory

#### 2.1. Discretization of the Closed Sound Field Using Plane-Wave-Enriched FEM

#### 2.2. Setup of the Plane-Wave Number for q-Refinement

## 3. Numerical Experiments

#### 3.1. Problem Description and Numerical Setup

#### 3.2. Effectiveness of the Variable q-Approach against the Constant q-Approach

#### 3.3. Mesh and Room Geometry Effects on the Robustness of the PW-FEM Using the Variable q-Approach

#### 3.4. Performance Comparison with Classical Linear and Quadratic FEMs

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An example of q-refinement using the Constant q- and Variable q-approach: (

**a**) the difference between Constant q and Variable q in the element patch composed of a 0.1 m × 0.1 m square element, two 0.5 m × 0.1 m rectangular elements, and a 0.5 m × 0.5 m square element; (

**b**) the change in the adding plane-wave number in the Variable q-approach at frequencies up to 4 kHz for the case with C = 8 when using ${h}_{\mathrm{max},i}^{\Omega}$ = 0.5 m and 0.1 m.

**Figure 2.**Two office room models with 21 sound-receiving points: (

**a**) office room with the rib-type acoustic diffuser (Model A) and (

**b**) more simply shaped office room without the acoustic diffuser (Model B). The room model includes three BCs: weakly absorbing surface ${\Gamma}_{\mathrm{z},1}$, frequency-dependent absorbing surface ${\Gamma}_{\mathrm{z},\mathrm{m}}$ assuming a honeycomb-backed MPP sound absorber, and vibration boundary ${\Gamma}_{\mathrm{v}}$ assuming a loudspeaker. The absorption characteristics of the MPP absorber are also shown.

**Figure 4.**Comparisons of the frequency responses at R1 among the PW-FEM using the Variable q-approach and the Constant q-approach and the reference solution: (

**a**) Mesh 1, (

**b**) Mesh 2, and (

**c**) Mesh 3.

**Figure 5.**A comparison of the SPL distribution at 1 kHz among (

**a**) the reference solution, (

**b**) the PW-FEM using the Constant q-approach, and (

**c**) the PW-FEM using the Variable q-approach.

**Figure 6.**A comparison of the absolute errors using the Variable q-approach and the Constant q-approach.

**Figure 7.**Comparisons of the absolute errors at eight frequencies as a function of C for Model A and Model B with three meshes, Mesh 1–Mesh 3: (

**a**) 63 Hz, (

**b**) 125 Hz, (

**c**) 250 Hz, (

**d**) 500 Hz, (

**e**) 1 kHz, (

**f**) 2 kHz, (

**g**) 3 kHz, and (

**h**) 4 kHz.

**Figure 8.**A comparison of the SPL distribution at 4 kHz of the reference solution and the PW-FEM using Mesh 2 with $C=9$ and $C=11$ and the SPL difference from the reference solution.

**Figure 9.**A comparison of the absolute errors of the PW-FEM using Mesh 1, Mesh 2, and Mesh 3 and the classical linear and quadratic FEMs.

**Figure 10.**Comparisons of the waveforms at R1 between (

**a**) the reference solution (Ref) and the PW-FEM (Mesh 3) at time t = 0–0.02 s, (

**b**) the Ref and the PW-FEM (Mesh 3) at time t = 0.3–0.32 s, (

**c**) the Ref and the quadratic FEM at time t = 0–0.02 s, and (

**d**) the Ref and the quadratic FEM at time t = 0.3–0.32 s.

**Figure 12.**Comparisons of the CPU times on sequential calculation of the PW-FEM with Mesh 1, Mesh 2, and Mesh 3 and the classical linear and quadratic FEMs: (

**a**) CPU times in constructing the global stiffness and mass matrix, $\mathit{K}$ and $\mathit{M}$, (

**b**) CPU times in solving the linear system of equations, and (

**c**) CPU times in total.

**Figure 13.**Comparisons of (

**a**) the DOFs, (

**b**) the number of nonzero elements, and (

**c**) the peak memory consumption, of the PW-FEM with Mesh 1, Mesh 2, and Mesh 3 and the classical linear and quadratic FEMs.

Model A | Model B | |||||
---|---|---|---|---|---|---|

20 Hz–1.5 kHz | 1.5–3 kHz | 3–4 kHz | 20 Hz–1.5 kHz | 1.5–3 kHz | 3–4 kHz | |

${N}_{\mathrm{element}}$ | 324,900 | 1,299,600 | 5,198,400 | 323,900 | 1,295,600 | 5,182,400 |

${N}_{\mathrm{node}}$ | 326,240 | 1,302,280 | 5,203,760 | 325,140 | 1,298,080 | 5,187,360 |

h | 0.01 m | 0.005 m | 0.0025 m | 0.01 m | 0.005 m | 0.0025 m |

${n}_{\mathrm{w}}$ | 0.0006–0.044 | 0.022–0.044 | 0.022–0.029 | 0.0006–0.044 | 0.022–0.044 | 0.022–0.029 |

Model A | Model B | |||||
---|---|---|---|---|---|---|

Mesh 1 | Mesh 2 | Mesh 3 | Mesh 1 | Mesh 2 | Mesh 3 | |

${N}_{\mathrm{element}}$ | 828 | 1302 | 3259 | 808 | 1282 | 3239 |

${N}_{\mathrm{node}}$ | 909 | 1399 | 3403 | 869 | 1359 | 3363 |

${\Omega}_{\mathrm{max},i}$ | 0.1–0.5 m | 0.1–0.25 m | 0.1 m | 0.1–0.5 m | 0.1–0.25 m | 0.1 m |

${n}_{\mathrm{w}}$ | 0.006–5.88 | 0.006–2.94 | 0.006–1.18 | 0.006–5.88 | 0.006–2.94 | 0.006–1.18 |

Model A | ||
---|---|---|

20 Hz–2 kHz | 2–4 kHz | |

${N}_{\mathrm{element}}$ | 51,984 | 207,936 |

${N}_{\mathrm{node}}$ | 209,008 | 833,888 |

h | 0.025 m | 0.0125 m |

${n}_{\mathrm{w}}$ | 0.001–0.15 | 0.074–0.15 |

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**MDPI and ACS Style**

Mukae, S.; Okuzono, T.; Sakagami, K. On the Robustness and Efficiency of the Plane-Wave-Enriched FEM with Variable *q*-Approach on the 2D Room Acoustics Problem. *Acoustics* **2022**, *4*, 53-73.
https://doi.org/10.3390/acoustics4010004

**AMA Style**

Mukae S, Okuzono T, Sakagami K. On the Robustness and Efficiency of the Plane-Wave-Enriched FEM with Variable *q*-Approach on the 2D Room Acoustics Problem. *Acoustics*. 2022; 4(1):53-73.
https://doi.org/10.3390/acoustics4010004

**Chicago/Turabian Style**

Mukae, Shunichi, Takeshi Okuzono, and Kimihiro Sakagami. 2022. "On the Robustness and Efficiency of the Plane-Wave-Enriched FEM with Variable *q*-Approach on the 2D Room Acoustics Problem" *Acoustics* 4, no. 1: 53-73.
https://doi.org/10.3390/acoustics4010004