A Petrov-Galerkin Natural Element Method Securing the Numerical Integration Accuracy

2-D large deformation analysis of nearly incompressible body by natural element method

“…These functions called Laplace interpolation function (or nonSibsonian interpolation function) [9,10] possess several useful properties like the Kronecker delta property, as Lagrange interpolation functions used in finite element method. Thanks to these properties, the essential boundary condition can be easily and accurately enforced [11,12], as in the finite element method. Meanwhile, the natural element method does not require extra effort to generate the background cell, because it utilize a set of Delaunay triangles, which are automatically identified in process of the basis function definition, as its background cell.…”

“…Sukumar et al [11,13] applied it to various problems in two-dimensional linear elastostatics to examine its accuracy and robustness, and they tried to couple finite element method with the natural element method. Cho and Lee [12] introduced a Petrov-Galerkin natural element method providing the best convergence rate in unconstrained two-dimensional linear elasticity problems in both convex and non-convex domains. Through the application to linear vibration and wave propagation problems, Bueche et al [14] confirmed that the overall performance of the natural element method in linear elastodynamics is better than the linear finite element method.…”

Satic states of cohesive granular media

“…The tensile contacts contribute negatively to the average pressure of the medium. The average pressure is given classically by (5) (6) a .. = (ria)…”

Polygon-wise constant curvature natural element approximation of Kirchhoff plate model

“…Here, the support of / I is defined as the intersection of the convex hull x CH ð@Þ (i.e. the problem domain x) and the union [cirðD IJK Þ of Delaunay circumcircles defined by the node x I and its neighbor nodes: The detailed discussion on the properties of Laplace interpolation functions may be referred to our previous paper (Cho and Lee, 2006a). As well, the behavior of Laplace interpolation functions corresponding to the boundary nodes and the adjacent interior nodes will be partially discussed in Section 4.3.…”

“…4. The reader may refer to Cho and Lee (2006a) for the detailed explanation of the numerical integration in the natural element method. On the other hand, the calculation of K I are straightforward once B I are line-integrated.…”