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

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.

[1]  Angelika Bunse-Gerstner,et al.  h2-norm optimal model reduction for large scale discrete dynamical MIMO systems , 2010, J. Comput. Appl. Math..

[2]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[3]  Mohammad Monir Uddin,et al.  Structure preserving model order reduction of large sparse second-order index-1 systems and application to a mechatronics model , 2016 .

[4]  Sarah Wyatt,et al.  Issues in Interpolatory Model Reduction: Inexact Solves, Second-order Systems and DAEs , 2012 .

[5]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[6]  Robert Skelton,et al.  Model reductions using a projection formulation , 1987, 26th IEEE Conference on Decision and Control.

[7]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[8]  Mohammad Monir Uddin.  Computational methods for model reduction of large-scale sparse structured descriptor systems , 2015 .

[9]  Paul Van Dooren,et al.  Model Reduction of MIMO Systems via Tangential Interpolation , 2005, SIAM J. Matrix Anal. Appl..

[10]  N. Martins,et al.  Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models , 2008, IEEE Transactions on Power Systems.

[11]  Mohammad Monir Uddin,et al.  Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation , 2012, Production Engineering.

[12]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[13]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[14]  Serkan Gugercin,et al.  Model Reduction of Descriptor Systems by Interpolatory Projection Methods , 2013, SIAM J. Sci. Comput..