Generalized splines and exact nodal solution by finite element method

Abstract

This paper discusses a method for constructing generalized splines, which is based on a matrix or structural interpretation of the mathematical theory on these functions. It is suggested throughout the paper that both the terminology and the methods of analysis in structures and strength of materials are very natural and, therefore, apt for this field of splines. The proposed method allows a wide range of splines to be addressed by means of a single and simple methodology: changing the characteristics of the spline from some subintervals to others (modification of weights, tension parameters, etc.), different conditions of interpolation in different nodes, etc. A noteworthy contribution is that new conditions of interpolation are considered, which are defined as individual actions (loads). Furthermore, it is found and shown that the solution of one-dimensional boundary value problems using the finite elements method is exact at the nodal points when certain spaces of finite dimension approximation engendered by generalized splines are used. The concept of equivalent action is developed as a generalization of the notion of equivalent nodal action (equivalent nodal loads). Finally, an illustration is given of how the developed methodology, based on the aforesaid matrix interpretation, can be applied, including examples of splines in the field of graphics, analysis of continuous beams on an elastic foundation, subjected to bending moment and tension or compression, and in dynamic problems.