Direct determination of finite element local smoothing matrices

In the finite element, local discrete smoothing of an exact least squares fit to the values of the stress at some sampling points is commonly employed. This seems to be the most suitable approach, since the displacement finite element procedure itself may be interpreted as a least squared error procedure. A matrix of shape functions can be evaluated at the sampling points with a particular coordinate transformation (N). The matrix having the reciprocal of such eigenvalues as characteristic values exhibits with the stated transformation the same form of the starting one. To extend the proof to any element, the spectral properties of the N matrices should be checked. The eigenvectors are ortho-normalized with respect to the unit matrix; as a consequence the outlined procedure can be regarded as a normalization to the unit matrix of the matrix of the shape functions calculated at the sampling points of an element, or interpreted as an extrapolation of the matrix N** minus **1 to some points, which have given co-ordinates ( xi prime , eta prime , zeta prime ) with reference to the original frame. 4 refs.