We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green's function by a small distance $h>0$. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter $h$ controls the conditioning of the discrete system. We provide mathematical guidance to choose $h$, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The results are compared to analytical solutions for canonical problems. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.

翻译：我们提出了一种新颖的虚拟源方法，用于构建外部波传播问题的边界积分方程。与经典边界积分公式不同，我们将格林函数的奇点位移一个小距离$h>0$。因此，离散化可以在具有连续核的实际物理边界上进行，从而任何简单的求积方案都可用来逼近积分算子。利用表面辐射条件，我们结合解的单层和双层势表示，在离散化后得到一个良态系统。虚拟位移参数$h$控制离散系统的条件数。我们提供了数学指导，根据波长和网格细化选择$h$，以在精度和稳定性之间取得平衡。展示了概念验证实现，包括二维和三维环境中的分段线性和等几何单元公式。我们观察到异常良好的谱特性，并使用无矩阵GMRES迭代求解相应系统。将结果与典型问题的解析解进行比较。我们得出结论，所提方法对于广泛的波长和网格细化能给出精确解和良好稳定性。