The scattering and transmission of harmonic acoustic waves at a penetrable material are commonly modelled by a set of Helmholtz equations. This system of partial differential equations can be rewritten into boundary integral equations defined at the surface of the objects and solved with the boundary element method (BEM). High frequencies or geometrical details require a fine surface mesh, which increases the number of degrees of freedom in the weak formulation. Then, matrix compression techniques need to be combined with iterative linear solvers to limit the computational footprint. Moreover, the convergence of the iterative linear solvers often depends on the frequency of the wave field and the objects' characteristic size. Here, the robust PMCHWT formulation is used to solve the acoustic transmission problem. An operator preconditioner based on on-surface radiation conditions (OSRC) is designed that yields frequency-robust convergence characteristics. Computational benchmarks compare the performance of this novel preconditioned formulation with other preconditioners and boundary integral formulations. The OSRC preconditioned PMCHWT formulation effectively simulates large-scale problems of engineering interest, such as focused ultrasound treatment of osteoid osteoma.
A domain decomposition preconditioning for the integral equation formulation of the inverse scattering problem
我们提出了区域分解预处理器的解决方案的积分方程制定的正、逆声散射问题的点散射体。我们研究了正问题和反问题，并提出了预处理技术，以加快迭代求解器。对于前向散射问题，我们扩展了{ \em“A restricted additive Schwarz preconditioner for general sparse线性系“，SIAM Journal on Scientific Computing，21（1999），pp. 792--797}，到积分方程。我们结合这个区域分解预条件与低秩校正，这是很容易构建，形成一个新的预条件。对于逆散射问题，我们使用的前向问题的预处理器作为一个积木构建一个预处理器的高斯-牛顿海森。我们提出的数值结果表明，这两个预处理策略的性能。
High-performance modeling acoustic and elastic waves using the Parallel Dichotomy Algorithm
Institute of Computational Mathematics and Mathematical Geophysics（计算数学与数学地球物理研究所）；Budker Institute of Nuclear Physics（布德刻核物理研究所）；Novosibirsk State University（新西伯利亚国立大学）
A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions
本文考虑一类具有Dirichlet、Neumann和阻抗型混合边界条件的声波传播问题。传输接口可以以与边界条件的位置无关的一般方式加入域边界。我们将推导出一个\textit{direct}，\textit{space-time retarded boundary integral equation}的公式，其中两个柯西数据都作为边界阻抗部分的未知数。这就需要定义的单迹空间，其中纳入齐次狄利克雷和诺依曼条件的相应部分的边界。利用拉普拉斯域上的算子演算技术证明了该方程的连续性和唯一性。
