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Distributions of sound pressure and numerical error inside the car cavity: (A) analytical solution, (B) LMFS results, and (C) absolute error. LMFS, localized method of fundamental solutions

声学反问题是物理力学中的一个重要研究领域，被广泛应用于声呐、医学成像、材料的无损探伤等领域。声学正问题是根据已知的声源或声场边界信息求解结构内部声场分布，反问题则是由声场或边界的部分信息确定未知边界信息、声源位置或声传播介质参数等。声学反问题不同于正问题，测量数据的某种微小扰动可能会导致反演结果的极大误差。传统方法仍面临数值不稳定、复杂结构网格划分、数值积分困难等问题，因此发展高效、稳定的新型数值算法用于复杂结构声学反演势在必行。

青岛大学王发杰副教授团队在《国际机械系统动力学学报（英文）》（International Journal of Mechanical System Dynamics, IJMSD）发表“基于无网格局部基本解法的三维复杂结构声学边界条件反演”研究论文。该文基于近年来提出的新型半解析无网格方法——局部基本解法，建立了一种简单、精确、高效的无网格数值模型，用于三维复杂结构声学边界条件的反演分析。局部基本解法具有数学形式简单、计算精度高、占用内存少等优势。它与传统方法不同，所形成的稀疏线性系统避免了病态问题，可采用常规解法进行求解。该算法的有效性和精确度经过含有高水平噪声的复杂结构数值试验得到了验证。该研究为复杂结构的声学反演提供了一种新的、简单高效的数值方法。

This paper proposes a semi-analytical and local meshless collocation method, the localized method of fundamental solutions (LMFS), to address three-dimensional (3D) acoustic inverse problems in complex domains. The proposed approach is a recently developed numerical scheme with potential of being mathematically simple, numerically accurate, and requiring less computational time and storage. In LMFS, an overdetermined sparse linear system is constructed by using the known data at the nodes on the accessible boundary and by making the remaining nodes satisfy the governing equation. In the numerical procedure, the pseudoinverse of a matrix is solved via the truncated singular value decomposition, and thus the regularization techniques are not needed in solving the resulting linear system with well-conditioned matrix. Numerical experiments, involving complicated geometry and high noise level, confirm the effectiveness and performance of LMFS for solving 3D acoustic inverse problems.

Keywords:

acoustic inverse problems, localized method of fundamental solutions (LMFS), 3D Helmholtz equation, meshless method, inverse Cauchy problem

DOI: 10.1002/msd2.12031

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https://onlinelibrary.wiley.com/doi/10.1002/msd2.12031

Cite this article: Chen Z, Wang F, Cheng S, Wu G. Localized MFS for three-dimensional acoustic inverse problems on complicated domains. Int J Mech Syst Dyn. 2022;2(1):143-152.

该文亮点：

1. 建立了基于局部基本解法的无网格数值模型，用于三维复杂结构声学边界条件的反演分析；

2. 该模型形成的稀疏线性系统相较于传统方法避免了病态问题，且易于进行大规模复杂结构求解；

3. 通过含高水平噪声的复杂结构数值试验证明了提出算法的有效性和精确度。

1. A semi-analytical and local meshless collocation method, the localized method of fundamental solutions (LMFS), is proposed to address 3D acoustic inverse problems in complex domains.

2. Compared with traditional methods, the sparse linear system formed by this scheme avoids ill-posed problems, and it can readily simulate large-scale structures.

3. Numerical experiments, involving complicated geometry and high noise level, confirm the effectiveness and performance of LMFS for solving 3D acoustic inverse problems.

作 者 简 介

陈增涛 青岛大学机电工程学院硕士研究生，主要从事汽车噪声、振动与声振粗糙度（NVH）性能的CAE仿真与分析，振动与噪声控制方面的研究工作。