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Abstract
The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.
Highlights
The finite elements with inter-elemental continuous differentiability are more complicated than those providing only continuity
As for single, noncomposite rectangles, the Bogner-Fox-Schmit (BFS) element [10] is the most popular and simplest one in the family of elements discribed by Zhang [11]. All these elements are widely used in the conforming finite element method for the biharmonic equation and other equations of the fourth order along with mixed statements of problems and a nonconforming approach [12,14,17]
A direct application of the BFS-element is restricted to the case of a simple domain that is composed of rectangles or parallelograms whose sides are parallel to two different straight lines
Summary
The finite elements with inter-elemental continuous differentiability are more complicated than those providing only continuity. A direct application of the BFS-element is restricted to the case of a simple domain that is composed of rectangles or parallelograms whose sides are parallel to two different straight lines This condition fails even in the case of a simple polygonal domain where the intersection of the boundary with rectangles results in triangular cells (cf Figure 1). One can construct the special triangulation compatible with the boundary as some isoparametric image [19] of a domain composed of rectangles with sides parallel to the axes This way requires solving some additional boundary value problems for the construction of such a mapping that is smooth over the whole domain. In this paper we suggest to use the BFS-elements in the direct way (without an isoparametric mapping) for a couple with the proposed triangular Hermite elements with 13 degrees of freedom These triangular elements supplement BFS-elements in the following sense.
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