A refinement-based triangular meshing algorithm is proposed to generate finite element meshes of arbitrary two-dimensional domains. The boundary is triangulated by the ear-removal method, constructing an initial mesh with coarse elements. To meet the pre-specified sizing requirement by refining the initial mesh, a novel mesh refinement method, Delaunay-optimized bisection, is presented. By combining global-longest-edge bisection with Delaunay edge swapping, well-shaped triangular elements are constructed. Meshing examples are presented along with mesh statistics, showing that the presented algorithm is capable of generating quality finite element meshes for arbitrary 2D domains.