Reduced-order modeling of index-1 vibrational systems using interpolatory projections

Abstract

Large sparse second order index-1 descriptor systems arise in various disciplines of science and engineering, such as constraint mechanics or multibody dynamics, mechatronics (where mechanical and electrical elements are coupled), but also RLC circuit design. Simulation, controller design and design optimization are only some applications of such models. Either of these tasks, just like any other many-query situation becomes unfeasible when the system is high dimensional. This paper discusses an algorithm to obtain a reduced state space model of a large sparse second order index-1 system using an interpolatory projection method based on the iterative rational Krylov algorithm (IRKA). In each iteration of this algorithm, we need to solve a number of linear systems. The main contribution of this paper is to solve these linear systems by exploiting the sparsity of the original model, which reduces the computational cost drastically. The algorithm is applied to a micro-mechanical piezo-actuated structural FEM model of a certain building block of a machine tool. Numerical experiments with a complex 3d model of an adaptive spindle support (a piezo-mechanical multiphysics system) show the effectivity and efficiency of the techniques.