High order transition elements: The xy-element concept—Part I: Statics

Abstract

Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element -xNy-element -which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h-and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.

Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions

“…At this point, only the basic idea is sketched and we refer to the pertinent literature [16][17][18][19][49][50][51]61,62] and the references cited therein. In particular, the authors recently presented a comprehensive analysis and generalization of this class of elements in [13]. The interested reader may refer to this publication for further details on the concepts outlined in the current work.…”

“…Similarly, by employing quadratic interpolation with evenly spaced nodes along all edges, we recover the well-known shape functions of the eight-node 'Serendipity' element. A graphical derivation of similar shape functions is given in [68], illustrating the more general concept developed in [13] for simple elements. By further increasing the polynomial degree, we can construct elements with an arbitrary interpolation order on the edges.…”

“…Using the above recipes, we can apply any combination of piece-wise interpo-lation of arbitrary order on each of the edges. Many examples and detailed analyses can be found in [13].…”

“…This concept also allows using a piecewise polynomial interpolation along any of the edges, such that elements of different sizes can be coupled. In a recent publication, we presented a detailed review and generalization of these classes of transition elements [13]. There, we refer to these general transition elements as xNy-elements to indicate that it is, in principle, possible to couple arbitrary element families.…”

“…2.6. Evaluating stresses is, however, not conceptually different from conventional finite elements, where stresses are discontinuous between adjacent elements[13]. Furthermore, to obtain stresses in an SBFE subdomain, the analytical solution in the ξ -direction has to be taken into account.…”