Adaptive Sparse Galerkin Methods for Vibrating Continuous Structures

Adaptive reduced-order methods are explored for simulating continuous vibrating structures. The Galerkin method is used to convert the governing partial differential equation (PDE) into a finite-dimensional system of ordinary differential equations (ODEs) whose solution approximates that of the original PDE. Sparse projections of the approximate ODE solution are then found at each integration time step by applying either the least absolute shrinkage and selection operator (lasso) or the optimal subset selection method. We apply the two projection schemes to the simulation of a vibrating Euler–Bernoulli beam subjected to nonlinear unilateral and bilateral spring forces. The subset selection approach is found to be superior for this application, as it generates a solution with similar sparsity but substantially lower error than the lasso.