TABLE OF CONTENTS Given the initial stiffness matrix K* in the decomposed form of Eq. (3.2) and the initial displacement vector r*, the problem considered in this chapter (see Section 3.1) is to estimate the modified displacements r due to changes in the design, without solving the complete set of modified analysis equations.
The Combined Approximations (CA) approach developed in this chapter is based on the integration of several concepts and methods, most of them presented in previous chapters. These include series expansion, reduced basis, matrix factorization and Gram-Schmidt orthonormalization. In the approach presented, the terms of the binomial series are used as high quality basis vectors in a reduced basis expression. The advantage is that efficient local approximations (series expansion) and accurate global approximations (the reduced basis method) are combined to achieve an effective solution procedure. Due to the nature of the selected basis vectors, high accuracy is often achieved by considering only a few vectors. Yet, the accuracy of the results can always be improved at the expense of more computational effort, by considering higher-order terms.
The main advantages of the CA approach, which can be studied in terms of several criteria, are as follows:
Generality. Various analysis models (linear, nonlinear, static, dynamic), different types of structures (trusses, frames, grillages, continuum structures) and all types of changes in the design (cross-sectional, geometrical, topological and material) may be considered. The changes can be of different extent, varying from changes in only a few variables to changes in...
