Keywords: Green's functions, functionals, finite elements, discretization error, modeling error, sensitivity.

A structure consists of infinitely many mechanisms - all bolted and fixed so that the structure can carry the load. But if we release one mechanism, apply a unit rotation or dislocation, then this will induce a movement in the structure and the work done by the load on acting through this movement is equal to M(x) . 1 or V(x) . 1, is equal to the Dirac energy [1]. In finite element (FE) analysis we hinder the movements of the structure and so the mechanism obtains an incorrect measure of how large the Dirac energy really is, and consequently M_h(x) not = M(x) and V_h(x) not = V(x). That is the kinematics of a mesh determines the accuracy of an FE solution

• mesh = kinematics = accuracy of influence functions = quality of results.

This is motivation to reduce the error G-G_h in the Green's functions. A second source of errors are the model parameters which too have an influence on the shape of the Green's function. So numerical accuracy (verification) and model accuracy (validation) hinge on an accurate approximation of the underlying Green's function.

A change in the bending stiffness EI in one frame element alone implies a global change in the Green's function and normally requires a full new evaluation cycle by integrating over all frame elements of the structure. We propose an alternative formulation where integration is done only over the frame element [x₁,x₂] that changes to assess the change in u(x) or sigma(x) at an arbitrary point x of the structure.

Normally the Green's functions decay with the distance r = |y-x| between the source point y and the field point x but each Green's function comes with its own domain of influence, its own mode of decay and these patterns - which are specific for each Green's function - encapsulate the design sensitivities of a structure. We suggest that questions such as: 'How accurate is the model and how sensitive is the output to changes in the parameters or how sensitive are the regions with high stress gradients to the model parameters?' should be answered by studying the Green's functions. The plots or charts of these functions are an ideal tool to weight the consequences which modifications of model parameters will have on the stress distribution in a structure.

We provide some numerical examples (bridge structures and plates) to demonstrate the efficiency and accuracy of the proposed method.

1

F. Hartmann, C. Katz, "Structural Analysis with Finite Elements", Springer-Verlag Berlin Heidelberg New York, 2nd Ed., 2007. doi:10.1007/s11012-005-6002-5

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