Discrete Gradient Finite Point Method

The objective of this work is the development of one point-cloud based numerical method that alleviates the difficulty of mesh generation of finite element method and the defects of classic meshfree methods. This new methodology, which combines the salient features of the discrete method and the Galerkin FEM, works directly on discrete gradient constructed over arbitrary nodes and thus bypasses meshing. Its formulations exist on a linearly exact discrete gradient operator, and automatically satisfy patch tests. The other elegant trait is that the computation of the discrete gradient computation involves only algebraic manipulations on vectors associated with the Voronoi diagram, and is very efficient once the tessellation is in place. It is obvious that this new method has the same advantage, which the approximation is constructed over local clusters without particular topological connection, as meshfree method has. Moreover, since the nodal values in this method correspond to the physical value of the dependent variable, all employed points can be considered as real physical points. As a result, the combination with finite element model or other numerical model becomes very convenient. In our numerical examples, this method displays a better accuracy than traditional FEM, and facile connectivity, efficiency as well, which is a big issue of meshfree method. Based on those standout traits, this method can be widely in structural analyses, especially in image processing because the methodology seamlessly connects to image database.