Statistical finite elements for vibration problems

Finite element (FE) methods remain one of the most widely used approaches across science and engineering for computing numerical solutions to partial differential equations. Given their widespread use, it is a natural family of methods to be considered within the framework of probabilistic numerics. Previous work has introduced the “statistical finite element method” (statFEM) as a tool for coherent treatment of uncertainty when working with FE models. Current formulations of statFEM rely on the discretised system being represented by a linear system of equations which is then solved. Within the scope of FE approaches there exists an alternative solution for cases where the modeller wishes to investigate the dynamic properties, i.e. resonant frequencies and associated mode shapes, of a system. Recovery of these properties requires solving a generalised eigenvalue problem utilising the discretised mass and stiffness matrices. The contribution of this work is to show how the approach of statFEM may be readily expanded to also cover this case by forming an approximate distribution over the eigenvalues and eigenvectors given a random field prior over one or more of the model properties chosen as a Gaussian process. We demonstrate the effectiveness of this approach on the classic test case of a cantilevered beam showing the approximated uncertainty recovered over both the eigenvalues and the eigenvectors.