# Department of Mechanics: Seminar: Abstract Krysl 2019

### Using a priori model reduction for the FEM to quickly approximate free-vibration response of solids

#### Petr Krysl, University of California, La Jolla, San Diego, USA

Modal expansion is a workhorse used in many engineering analysis algorithms. One example is the coupled boundary element-ﬁnite element computation of the backscattering target strength of underwater elastic objects. To obtain the modal basis, a free-vibration (generalized eigenvalue) problem needs to be solved. This tends to be expensive when there are many basis vectors to compute. In the above mentioned backscattering example it could be many hundreds or thousands. Excellent algorithms exist to solve the free-vibration problem, and most use some form of the Rayleigh-Ritz (RR) procedure. The key to an eﬃcient RR application is a low-cost transformation into a reduced basis. In this work we show how a cheap a prior transformation can be constructed for solid-mechanics ﬁnite element models based on the notion of coherent nodal clusters. The inexpensive RR procedure leads to not insigniﬁcant speedups of the computation of an approximate solution to the free vibration problem.