The quasi-uniformity condition for reproducing kernel element method meshes

Abstract

The reproducing kernel element method is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-δ property. To achieve these properties, the underlying mesh must meet certain regularity constraints. This paper develops a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. The algorithm is demonstrated on several mesh types. Finally, a guide to generation of quasi-uniform meshes is discussed.

The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers

“…Moreover, we will also consider different ordering algorithms (including METIS [30] and AMD [31]) to illustrate that the main trends are also independent of the selected ordering, provided that this ordering algorithm is competitive. While this work chooses to focus on isogeometric methods, the results apply to a broader class of hpk-finite elements, such as those proposed by Liu et al [32][33][34][35][36].…”

Symmetric global partition polynomials for reproducing kernel elements

“…Furthermore, the RKEM interpolants are globally smooth functions and therefore, there is no need to apply smoothing techniques to the solution. In all the examples, we have enforced the Kronecker-δ property only on the boundary nodes, using the concept of nodal isolation presented in [4]. Furthermore, a fourth-order conical window function with a radial support has been used to calculate the kernel and 36 Gauss points per element were used to integrate the weak form, in contrast to the 576 Gauss points per element used before.…”

“…where 1 , t e,1 ), ,x (s e,1 , t e,1 ), ,y (s e,1 , t e,1 ), · · · , (s e,2 , t e,2 ), ,x (s e,2 , t e,2 ), ,y (s e,2 , t e,2 ), · · · , (s e, 3 , t e,3 ), ,x (s e, 3 , t e,3 ), ,y (s e, 3 , t e,3 ) · · · (s e, 4 , t e,4 ), ,x (s e, 4 , t e,4 ), ,y (s e, 4 , t e,4 ) · · ·…”

Reproducing Kernel Element Method for Galerkin Solution of Elastostatic Problems

“…In this work we have used the circular domains for all the computation. In RKEM the radius of the circle can not be arbitrary, it need to be a function of the mesh [10]. The scheme for the circular domain is shown in Fig.…”