Calcolo della norma H-infinito per matrici di trasferimento relative alla risposta sismica di strutture in cemento armato

In this paper we propose three different algorithms for the computation of the H-infinity norm of transfer function matrix for linear dynamical systems. In particular, we focus our attention on systems created by concrete structures forced by a seismic base acceleration. The H-infinity norm represents the maximum structural response of these systems. The knowledge of this parameter is important in order to determine a rational design approach capable to minimize the structural response. It is also really important we understand what a transfer function matrix represents for structural systems and what role it plays in this process. In practice it is a function that links the input to the output of a dynamical system. Working on the equations that are set in frequency domain and not in period domain as it is usual for a structural dynamical analysis, we can easily understand that the transfer matrix relates input and output thanks to four matrices depending on the mass, stiffness and damping (thus the frequency) of the structures. First, two numerical investigations of one-storey one-bay frame and two-storey one-bay frame are performed in order to understand exactly how to create a transfer function matrix and how it works. Secondly, the finite element method is introduced to create the transfer matrix for big structures with lots of degrees of freedom. Then we implement three algorithms (namely Bisection, Granso and Newton) with MATLAB to compute the H-infinity norm that work as follows. The first algorithm is based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related Hamiltonian Matrix. The second and the third algorithms are based on theorems about matrix differentiation rules for eigenvalues and eigenvectors. For all algorithms three different numerical method are implemented in order to obtain the maximum value of H-infinity, respectively: Bisection method for the first one, GRANSO method for the second one and Newton's method for the last one. Finally, thanks to an investigation of 12 different types of structures, we demonstrate that the GRANSO algorithm is several times faster than the existing method (MATLAB's getPeakGain) but not very accurate for large-scale systems. On the other hand Newton algorithm is faster and accurate for any type of dynamical system.