This paper proposes a new Helmholtz decomposition based windowed Green function (HD-WGF) method for solving the time-harmonic elastic scattering problems on a half-space with Dirichlet boundary conditions in both 2D and 3D. The Helmholtz decomposition is applied to separate the pressure and shear waves, which satisfy the Helmholtz and Helmholtz/Maxwell equations, respectively, and the corresponding boundary integral equations of type $(\mathbb{I}+\mathbb{T})\bs\phi=\bs f$, that couple these two waves on the unbounded surface, are derived based on the free-space fundamental solution of Helmholtz equation. This approach avoids the treatment of the complex elastic displacement tensor and traction operator that involved in the classical integral equation method for elastic problems. Then a smooth ``slow-rise'' windowing function is introduced to truncate the boundary integral equations and a ``correction'' strategy is proposed to ensure the uniformly fast convergence for all incident angles of plane incidence. Numerical experiments for both two and three dimensional problems are presented to demonstrate the accuracy and efficiency of the proposed method.

翻译：本文提出了一种新的基于亥姆霍兹分解的窗函数格林函数（HD-WGF）方法，用于求解二维和三维情形下具有迪利克雷边界条件的时谐半空间弹性散射问题。通过应用亥姆霍兹分解分别处理压力波和剪切波，二者分别满足亥姆霍兹方程和亥姆霍兹/麦克斯韦方程组，并基于亥姆霍兹方程的自由空间基本解推导出耦合这两种波在无界表面上的边界积分方程组$(\mathbb{I}+\mathbb{T})\bs\phi=\bs f$。该方法避免了经典弹性问题积分方程方法中涉及的复杂弹性位移张量和牵引力算子的处理。随后引入一种光滑的“慢升”窗函数来截断边界积分方程组，并提出一种“修正”策略，以确保对所有平面入射角度的均匀快速收敛。文中给出了二维和三维问题的数值实验，以证明所提方法的精度和效率。