Keywords: theory, finite elements, vector equations, spherical shell.

This paper deals with a theory of generating finite elements based on differential equations. A vector is regarded to a synonym of an ordered set of quantities. On the basis of this definition we can use the concept of complement vectors similarly as in the set theory. Further we specify complement matrices as coeffient matrices of the complement vectors. The sets under consideration are the nodal quantities and their equations of a finite element. In addition we assume we have a differential equation or a system of them for our problem and know the solution or at least, can estimate or approximate it.

Using these assumptions a theory of six vector equations valid for stiffness, mixed and flexibility methods is presented. A first equation is used to generate finite elements and a second for handling the interior of the element. The former one determines the interaction between the vectors a and b which are complements of each other (as their coefficient matrices A and B). The physical meaning of this interaction equation depends on the content of its vectors.

As applications we consider axisymmetric spherical shell examples. One of the examples handles an open shell and two closed ones. The analytical solution is based on the theory of thin elastical shells. The solution of differential equations is exact it uses power series named as north and south pole functions. The solution is presented by Outinen [1] and also very briefly in the present paper. The open shell is also analyzed using a numerical method based on the use of finite differences. The results of calculating analytically the closed shells are compared with the ones published earlier. Some conclusions are presented on the basis of the theory and the examples.

1

Outinen H., "Isotrooppisen pallokuoren rotaatiosymmetrinen statiikka", Tampere University of Technology, Department of Mechanical Engineering, Applied Mechanics, Tampere, 1985.

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