## Abstract

This study makes the first attempt to extend the meshless boundary-discretization singular boundary method (SBM) with time-dependent fundamental solution to two-dimensional and three-dimensional scalar wave equation upon Dirichlet boundary condition. The two empirical formulas are also proposed to determine the source intensity factors. In 2D problems, the fundamental solution integrating along with time is applied. In 3D problems, a time-successive evaluation approach without complicated mathematical transform is proposed. Numerical investigations show that the present SBM methodology produces the accurate results for 2D and 3D time-dependent wave problems with varied velocities *c* and wave numbers *k*.

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## Acknowledgments

The work described in this paper was supported by the National Science Funds of China (Grant Nos. 11302069, 11372097, 11572111), the Chinese Postdoctoral Science Foundation (Grant Nos.2014M561565, 2015T80492), and the Foundation for Open Project of the State Key Laboratory of Acoustics (Grant No. SKLA201509), the Foundation for Open Project of the key laboratory of road construction technology and equipment, ministry of education (Grant No. 310825151132), the 111 Project (Grant No. B12032).

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## Appendix: The detailed derivation of Eq. (13)

### Appendix: The detailed derivation of Eq. (13)

The derivation of Eq. (13) will be provided via a circle domain with uniformly distributed points. It is beyond our capability to derive this formula in more general situations. However, our numerical experiments show that Eq. (13) can be extended to irregular domains with non-equally distributed points.

Consider a 2D Laplace equation on Dirichlet boundary condition

where \(\nabla ^{2}\) denotes the Laplacian operator, *u*(*x*) represents the potentials in domain \(\Omega \), \(\bar{{u}}(x)\) is the known function, \(\Gamma _D \) represents the Dirichlet boundary conditions.

The formulation of the SBM can be stated as

where \(G_0 (x_m ,s_n )=\ln (r_{mn} )\) is the fundamental solution of 2D Laplace equations, \(U_0^m \) the source intensity factors, \(\alpha _n \) the unknown coefficient.

Consider \(\Omega \) a circle with radius R. \(\left\{ {x_m } \right\} _{m=1}^N \) denote the collocation and source points uniformly placed on the boundary \(\partial \Omega \). \(L_n \) represents the length of the auxiliary line in 2D as shown in Fig. 5.

For \(x_m \in \partial \Omega \), it is easy to verify that the following contour integral is zero.

where *r* is the distance between \(x_m \) and source points *s* on the circle. Eq. (34) can be discretized as

Due to the symmetry property of the circle, the derivation is independent of the indices of the collocation points. \(m=1\) is used in the following derivation. Hence, we have

where \(x_1 =(1,0)\). Notice that

Substituting Eq. (37) into Eq. (36), and define the desingularized value of the natural logarithm function as *a*,

The key issue is to determine the value of \(\prod \nolimits _{n=2}^N {\sin \left( \frac{n-1}{N}\pi \right) } \). Let \(\omega =\cos (2\pi /N)+i\sin (2\pi /N)=\exp (2\pi i/N)\). Note that \(\omega \) is a solution of the equation

Similarly, \(\omega ^{2},\omega ^{3},...,\omega ^{N-1}\) are also the solution of Eq. (39). After some algebraic manipulations, Eq. (39) is equivalent to

For \(z=1\), we obtain

It follows that

Since

We have

Therefore,

From Eqs. (38) and (45), we have

It follows that \(U_0^m \) for the natural logarithm function is

where \(L_m =2\pi R/N\).

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Chen, W., Li, J. & Fu, Z. Singular boundary method using time-dependent fundamental solution for scalar wave equations.
*Comput Mech* **58**, 717–730 (2016). https://doi.org/10.1007/s00466-016-1313-z

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DOI: https://doi.org/10.1007/s00466-016-1313-z