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

This paper proposes a fast meshless scheme for acoustic sensitivity analysis by using the Burton–Miller-type singular boundary method (BM-SBM) and recursive skeletonization factorization (RSF). The Burton–Miller formulation was adopted to circumvent the fictitious frequency that occurs in external acoustic analysis, and then the direct differentiation method was used to obtain the sensitivity of sound pressure to design variables. More importantly, RSF was employed to solve the resultant linear system obtained by the BM-SBM. RSF is a fast direct factorization technique based on multilevel matrix compression, which allows fast factorization and application of the inverse in solving dense matrices. Firstly, the BM-SBM is a boundary-type collocation method that is a straightforward and accurate scheme owing to the use of the fundamental solution. Secondly, the introduction of the fast solver can effectively reduce the requirement of computer memory and increase the calculation scale compared to the conventional BM-SBM. Three numerical examples including two- and three-dimensional geometries indicate the precision and efficiency of the proposed fast numerical technique for acoustic design sensitivity analysis associated with large-scale and complicated structures.