  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.
