Abstract:In natural element method(NEM),the trial and test function are constructed with the natural neighbor interpolation(NNT) method. The interpolation is constructed with respect to the Voronoi tessellation of the scattered nodes in the problem domain. The integration of the weak form is performed in the Delaunay triangles,which is the dual diagram of the Voronoi tessellation when the Galerkin method is used to form the discrete system equation. But the integration domain does not align with the compacted support of the NEM shape function,which results in considerable integration error. The partition of unity quadrature method which employs the fact that the meshless shape function possesses the partition of unity is presented,and the integration can be done within the compacted support of the shape function. However,the shape and the size of NEM shape function are determined by the Voronoi tessellation and are somewhat complex. Therefore,the implementation of partition of unity quadrature in NEM involves the decomposition and mapping the support of NEM shape function. A single point quadrature rule is proposed based on the average strain of each Delaunay triangle,the average strain can be calculated by shift surface integral to contour integral via the divergence theorem,thus the derivatives of the shape function are unnecessary to form the global stiffness matrix. Numerical examples show both the accuracy and efficiency are improved. Considering both the accuracy and the efficiency,the point quadrature based on the average strain of Delaunay triangle is a better selection,though its convergence and accuracy are a little lower than the partition of unity quadrature method.