The reanalysis problem considered in this study is to find efficient and accurate approximations of the displacements r for various changes in a structure, without solving the complete set of modified equations Kr=R. In this formulation K is the nxn modified (positive definite) stiffness matrix, R is the given load vector and n is the number of degrees of freedom. It is assumed that the initial values R_0 and K_0 are given from exact analysis of the initial structure. The Combined Approximations (CA) method developed recently is an efficient reanalysis approach providing accurate results. The high accuracy of the results achieved by the method has been demonstrated in previous studies [e.g. 1]. Accurate solutions have been achieved efficiently in various cases of very large changes in the design, but the reason of the high quality of the results was not fully understood. The Conjugate Gradient (CG) method is an iterative method that can be used to solve the modified analysis equations. For the problem under consideration the convergence of the method might be slow. To accelerate the convergence rate it is possible to transform the problem such that the eigenvalue distribution of the coefficients matrix is improved. The convergence rate of the resulting Preconditioned Conjugate Gradient (PCG) method depends on the eigenvalues of the preconditioned matrix. It is proposed in this study to select the preconditioned matrix in such a way that the PCG method presented and the CA method provide theoretically identical results. Consequently, available results from one method can be applied to the other method. In particular, effective solution procedures developed in the past for the CA method can be used for the PCG method. In addition, various criteria and error bounds developed for CG methods can be used for the CA method.
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