According to the basic problem of the sub domains decomposition of simple polygon in the computational geometry and the finite element mesh generation of computational mechanics

an algorithm for decomposing a simple polygon into set of triangles and convex quadrilaterals is proposed in this paper in which rules of decomposition are developed. And these rules are propitious to the finite element mesh generation. The algorithm is guided by a weighting function for pairs of visible vertexes of the given polygon and the estimation factor of shape quality. The created weighting functions are related with the concave (convex) angles

so the function values are different for any two different concave (convex) angles. The algorithm is composed of three steps: (1) Decomposing the simple polygon into set of convex polygons step by step

(2) Decomposing the over six convex polygons into set of under six ones (include six ones)

(3). Decomposing the six and five convex polygons into set of triangles and convex quadrilaterals.