PHYSICALLY ADEQUATE CONDENSATION AND MIXED MODELS OF SERENDIPITY ELEMENTS

Abstract

Mixed serendipity models are models of finite elements with interpolant, which is represented by polynomials of different degree on each of two coordinates. Usage of such elements allow to coordinate elements of lower order in spheres, where sharp change of characteristics is not envisaged with elements of higher order in other spheres. The serendipity version of quadraticaly-cubic interpolation on the canonical square (|x| ≤ 1, |y| ≤ 1) is considered in the work. In the direction of 0x axis the function changes by the law of cubic parabola, in the direction of 0y it changes by the law of quadratic parabola. Lagrange prototype of such element has 12 nodes (two internal). As it is known internal nodes should be turned off in order to get serendipity model. The traditional procedure of condensation (reduction) involves making and solving the system of linear algebraic equations with 12×12 matrix. Then to eliminate internal nodes one should find the ’recipe’ of condensation, that is to build linear dependence of internal parameters (two) on boundary ones. Known examples show that mathematically grounded ‘recipe’ of condensation does not guarantee physical adequateness of variety of nodal loads of serendipity models. It happened to biquadratic element (Jordan ‘recipe’, 1970) and triangle of third order (Ciarlet-Raviart ‘recipe’, 1972). In serendipity models loads of angular nodes are negative both on to biquadratic element and bicubic one. It is surprising that mathematically grounded and elegant Tailor method proves this peculiarity of standard serendipity models. To avoid anomalies in the spectrum of nodal loads, one should start with building of the desired spectrum. It is the inverse problem, when one chooses desired integral characteristics first and after that identifies the basis which implements these characteristics. This particular ‘non-matrix’ approach is given in the work. The important peculiarity of non-matrix reduction lies in the fact that it excludes internal nodes, but keeps internal parameters. Availability of ‘hidden’ parameters allows to direct the formation of alternative serendipity surfaces. Meaningful opportunities of suggested approach allow always to get natural (physically adequate) spectrum of nodal loads. It is relevant to serendipity models.