## Dynamic structural monitoring, stiffness reconstruction and the underlying theory of the inverse problems in structural dynamics. Introduction.

Seminarium: **Dynamic structural monitoring, stiffness reconstruction and the underlying theory of the inverse problems in structural dynamics**

Z. Zembaty - **Introduction**

Contrary to well known forward problems in mathematics & engineering, in which one solves a differential equation or system of differential equations, in the inverse problems the question is about finding the parameters involved in these equations, basing on the input or output functions. This means that it is the model which is in question, not its predictions. That is why the inverse problems are often formulated as questions how to convert observed measurements into information about a physical object or a system.

A bunch of interesting, classic inverse problems can be found in geophysics. For example: how to infer about the Earth's interior basing on the earthquake source model (input), wave propagations (the model) and surface vibrations measurements (the output).

The inverse problems approaches are used extensively in climatology, weather predictions, oceanography, radar echo analyses, image processing, in constructing computational models of oil reservoirs as well as in the structural health monitoring.

Actually, general domain of inverse methods is covered by three major international journals:

- Inverse Problems
- Journal of Inverse and Ill-posed Problems
- Inverse Problems in Science and Engineering

and many more which are dominated by subjects related to inverse problems. Numerous monographs have been written about the inverse methods. When it comes to general inverse formulation a book 'Functional Analysis - Applications in Mechanics and Inverse Problems' by Lebedev et al. (2002) is worth mentioning. Structural dynamic problems are treated in an interesting way in the book by Gladwell entitled 'Inverse Problems in Vibration' (2004), while numeric approaches to solve the inverse problems can be found in the book 'Computational inverse techniques in nondestructive evaluation' by Liu & Chen (2003).

The inverse problem attracts attention of many scientists and engineers because it is a vital problem in building models of mathematical physics and engineering. However, the mathematical methodology is still not well developed and in most of the cases the solutions are not unique. So a detailed physical or engineering knowledge is necessary to obtain any meaningful results.

The subject of the Seminar of the Section of Computational Methods and Optimization of the Committee of Mechanics of the Polish Academy of Sciences is formulated as Dynamic structural monitoring, stiffness reconstruction and the underlying theory of the inverse problems in structural dynamics. It covers only selected approaches in the area of structural dynamics in civil & mechanical engineering. This area is known as Structural Health Monitoring (SHM).

The purpose of the introductory lecture is to bring to the audience a concise presentation on the wide spectrum of inverse methods and to present its simple formulation in application to structural dynamics