The finite-element method can easily handle complicated geometries because of the application of unstructured meshes. Unlike the Cartesian grid used in the finite-difference method, the unstructured mesh can follow the sharp interfaces that separate two layers of different properties. Therefore, the finite-element method can provide more accurate solutions for the simulation of seismic wave propagation. Meshes of good quality are required for the finite-element simulation. However, it is not trivial to set up an appropriate mesh. First
of all, the mesh should contain elements of good shapes and sizes. In addition, the sharp interfaces should coincide with the edges of the elements instead of intersecting with them. These requirements are formulated as an optimization problem with three terms, measuring the difference between the actual and prescribed scaling field, shape quality, and the area between prescribed curves and the nearest triangle edges. The solution of the optimization problem should provide the desired mesh. The mesh generator MESH2D was applied to obtain an initial mesh. The Matlab function minFunc was used to search for the minimum of the constructed objective function. Three weights balance the three terms in the objective function. When it comes to complicated models, these weights have to be chosen carefully to produce a reasonable mesh.