When we want to use reference points located arbitrarily in two- or three-dimensional regions, it is essentially difficult to design some structure-preserving methods. The reason is that we should discretize some Gauss-Green formulae keeping some mathematical properties in that situations. Based on Voronoi-Delaunay triangulations, we can find some beautiful discrete Gauss-Green formulae and apply them to design some structure-preserving numerical methods. In the talk, we will indicate those formulae and their proofs in detail and the obtained discrete variational derivative methods based on Voronoi cells. Furthermore, if it is possible, we will show some relaxed structure-preserving methods to decrease computation cost.