Multiple model updating using the finite element method over a polynomial algebra

Vibration control is an important issue when it comes to preserving the structural integrity of mechanical structures, particularly in the case of lightweight structures found aboard flying vehicles, such as printed circuit boards. In order for a simulation to be effective, a suitable numerical model of the structure must be used. Such a model can be obtained using a model updating method, such as that which is presented in this work, based on the modified constitutive relation error principle. However, like many other model updating strategies, this method can be very expensive. When the structural parameters change over time, it would be too time consuming to perform another full calculation. In order to circumvent this problem, we introduce a new technique called the 'finite element method over a polynomial algebra', which involves modules defined over a ring of truncated polynomials in multiple variables. In this paper, we illustrate the method with the updating and reduction of a model of a beam instrumented with piezoelectric sensors and actuators, which are all taken into account in the numerical model. This is a simple problem, yet it is representative of the updating of printed circuit boards