  In this paper, we propose a new nonuniform mesh method to simulate acoustic
scattering problems in two dimensional periodic structures with non-periodic
incident fields numerically. As existing methods are difficult to extend to
higher dimensions, we have designed the new method with such extensions in
mind. With the help of the Floquet-Bloch transform, the solution to the
original scattering problem is written as an integral of a family of
quasi-periodic problems. These are defined in bounded domains for each value of
the Floquet parameter which varies in a bounded interval. The key step in our
method is the numerical approximation of the integral by a quadrature rule
adapted to the regularity of the family of quasi-periodic solutions. We design
a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval.
We prove that the numerical method converges exponentially with respect to both
the number of subintervals and the number of Gaussian quadrature points. Some
numerical experiments are provided to illustrate the results.
