Keywords: finite elements, unified, theory, differential equations, vector equations, simplicity, brevity.

The paper deals with a unified theory to generate finite elements based on differential equations. A vector is regarded to be a synonym of an ordered set of quantities. On the base of this definition we can use the concept of complement vectors similarly as in set theory. Further we specify complement matrices as coefficient matrices of complement vectors. The sets under consideration are the nodal quantities and their equations. 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 guess or approximate it.

Using these assumptions a theory of vector equations valid for stiffness, mixed and flexibility methods is presented. One of the equations is used to generate finite elements and another one 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 inter-action equation depends on the content of its vectors. If matrix A is singular, vectors in the equation have to be reduced.

As an application of the theory the stretching state of a constant stress triangular plate element is considered. The transfer matrix and the loading vector in the direction of x-coordinate are determined for the plate. The problem is relevant in connection with the finite element transfer matrix method (FETM-Method) presented in reference [1].

The mathematical background of the theory consists of six vector equations. Using assumed functions as a basis for a finite element means that the solution functions of the homogenous differential equation (or a system of them) are replaced with these functions. The functions must be linearly independent, i.e. the inverse A^-1 exists. From the viewpoint of differential equations generation of a finite element is a problem where boundary values are given at nodal points as components of the vector a.

The determination of the loading (forcing) vector for use in the interaction equation is problematic, when handling multi-noded (more than two nodes) elements.

Quite a philosophical result is the equality of the nodal quantities with regard to the mathematics. All are components of vectors a and b.

The use of sets makes it possible to apply the theory with different types of elements. The term interaction equation describes naturally the relation between vectors a and b and in this context the term 'interaction matrix' is well suited.

1

Tesar, A. and Fillo L'u., "Transfer matrix method", Kluwer Academic Publishers, Dordrecht/Boston/London, 1988.

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