Geometry representation, modification and iterative design using RKEM

Symmetric global partition polynomials for reproducing kernel elements

“…The theory behind volume representation is found in [15] and it is explain briefly here. Given a boundary point set, find a perpendicular projection of the point onto the nearest RKEM boundary edge.…”

“…On the other hand, it is very desirable, from the meshing point of view, to restrict all vertex nodes to have the same DOF's, consequently the global to local numbering indexing do not require any special Solid circle means that a function value is interpolated and circle about a point means that both first derivatives are to be interpolated at that point algorithm. In [16,17] and [15], the DOF's mismatch problem was solved by selecting interpolations that provided equal or greater total DOF's required for the desired degree polynomial through additional derivative degrees of freedom. Use of derivative degrees of freedom leads to an added restriction for the successful use of a parametric formulation using a parent element.…”

“…The competitive advantage of RKEM is the easiness of enforcing the Kronecker-δ property, yet maintaining an arbitrary order and continuity of the shape functions. However, the triangular elements developed in [16,17] and tetrahedral element in [15] are not truly minimal, in that the necessary number of degrees of freedom per element exceed the number necessary to interpolate the complete polynomial reproduced by the element. Some number of higher-degree monomial terms appear in the global partition polynomials that cause some issues, to be discussed in Sect.…”

The quasi-uniformity condition for reproducing kernel element method meshes

“…In previous work, RKEM has been used to solve Galerkin weak form problems [7][8][9]11,14] and also isogeometric problems in [15]. However, the underlying requirement of quasi-uniformity has not received as much attention, briefly discussed in [8,13].…”

“…While analysis is an important topic, this paper is concerned with detailing the condition necessary for RKEM interpolant construction. This is critical for the use of RKEM in applications where it is used for other interpolation tasks, such as geometry representation [15].…”