# Finite Elements and Green’s Functions

• Friedel Hartmann
Chapter

## Abstract

In FE-analysis we substitute for the exact solution of the equation L u = p an approximation uh which is the exact solution of the equation L uh = ph. The special nature of the FE-solution allows to extend Betti’s Theorem (p1,u2) = (p2,u1) to the FE-solutions in the following sense (p1,uh 2,p2,u1 h) which establishes that the FE-solution is the scalar product of the approximate Green’s function Gh[x] and the original right-hand side p, namely uh(x) = (Gh[x],p). We must distinguish between weak and strong influence functions. Weak influence functions are based on the principle of virtual forces (Green’s first identity) and can only be formulated for displacement terms while strong influence functions, which can be formulated for displacement and force terms, are based on Betti’s theorem (Green’s second identity). In FE-analysis this distinction gets lost because of the approximate nature of the kernel functions. Influence functions in FE-analysis take a nodal form, that is they are evaluated by summing over the nodes (scalar product of two nodal vectors g and f or u and j). The columns of the inverse stiffness matrix are the nodal values of the Green’s functions. Infinite stresses pose a problem for influence functions because they imply that the kernel functions become unmeasurable when the dislocations, which trigger the Green’s functions, move to the singularity. The essential features of discrete FE-Green’s functions also apply in the case of mixed problems and the sensitivity of the p-method with regard to point sources also follows from the nature of the Green’s functions.

## Keywords

Shape Function Stiffness Matrix Nodal Displacement Influence Function Nodal Force
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