Abstract: A coupled-channel resonating group equation for orthogonal channel spaces is derived. It follows from the common resonating group equation by a recursion relation. The recursion extracts from higher channels all overlaps with lower channels, such that the higher channels become corrections to the lower ones. A physically meaningful definition of elimination potentials becomes possible. The new coupled channel resonating group equation allows the derivation ofphysical effective potentials by eliminating small corrections, only. It also allows the derivation oftechnical potentials, i.e. potentials with an unphysical off-shell behaviour, when the dominant part of the equation is eliminated. A numerical example demonstrates that linear dependence of the test function space is not harmful to the new equation.

Abstract: The electron correlation theories for extended systems require an orthogonal and at the same time well-localized virtual orbital system. An iterative method is suggested, by which extremely nonorthogonal basis sets can be orthogonalized without destroying the localization, in contrast to other well-known procedures. With the help of a general formulation of the problem not only the localization but other properties can be achieved as well. The method is compared to Lowdin's orthogonalization. Calculations for model and real systems were carried out and the convergence properties and the stability of the fixed points of the iterative procedure were investigated.

Abstract: The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

Abstract: The idea behind the well known conjugate gradient procedure is to solve a series of one-dimensional optimization problems along direction vectors that are a function of both the current gradient vector and the previous search vector. The search vectors are sequentially generated allowing the optimization process to move along one direction at a time while making these vectors Q orthogonal, where Q is the nxn weighting matrix from the quadratic objective function. In this work, a method is presented in which the search directions are all initially fixed as the columns of the Q matrix. It is then shown that for this choice, the Gram-Schmidt orthogonalization process can be used to locate the extremum in n steps. It is also shown that the original search directions become conjugate directions after these n steps. The net result is a new and efficient conjugate direction method.

Abstract: A radar doppler processor, comprising M, M=N-1, tap delay lines; N digital multipliers; a N-point fast fourier transform network; and a fast orthogonalizing network to orthogonalize each subband output signal to eliminate cross-correlations between all output signals.

Abstract: Orthogonal curvilinear co-ordinates represent the logical choice for description of the middle surface of the majority of shells in engineering practice. By means of orthogonalization of originally skew parameter lines at integration points for shell analysis by the finite element method (FEM), the remaining minority can be treated without having to resort to skew curvilinear co-ordinates. Therefore, it does not seem to be worth while to extend existing computer codes for shell analysis by the FEM, restricted to orthogonal curvilinear co-ordinates, to general curvilinear co-ordinates. Depending on the structure of such a code, this extension may require a substantial amount of recoding. (However, it is advisable to consider general curvilinear co-ordinates if a new computer code for shell analysis by the FEM, based on thin- or thick-shell theory, is written). Based on these considerations, the concept of ‘discrete orthogonalization of parameter lines’ has been developed. It is presented in this paper and applied to the analysis of a hyperbolic paraboloid groined vault subjected to dead load. In the numerical investigation it is demonstrated that treating skew parameter lines incorrectly as orthogonal has a significant effect on the results for the displacements and the internal forces.