A framework for parametric reduction in large-scale nonlinear dynamical systems
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ORIGINAL PAPER
A framework for parametric reduction in large-scale nonlinear dynamical systems Danish Rafiq
· Mohammad Abid Bazaz
Received: 6 March 2020 / Accepted: 18 September 2020 © Springer Nature B.V. 2020
Abstract This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.
1 Introduction and problem formulation Large-scale dynamical systems are often encountered across all engineering disciplines [1]. For instance, modelling partial differential equations (PDEs) via Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11071-020-05970-3) contains supplementary material, which is available to authorized users. D. Rafiq (B)· M. A. Bazaz Department of Electrical Engineering, National Institute of Technology, Srinagar 190006, India e-mail: [email protected] M. A. Bazaz e-mail: [email protected]
finite elements (FE) or finite difference (FD) usually produces a large-system of ordinary differential equations (ODEs) and to carry out efficient design or control, repeated simulations of such large FE models are required. This increases the associated computational cost and can become impractical at times. MOR aims at approximating such large-scale/complex full order models (FOMs) by reduced/simpler surrogate models sharing similar properties like stability, passivity, etc, while keeping the input–output relationship intact [1]. Methods like Krylov subspace projection [2,3], balanced truncation [4], modal reduction [6,7] are some of the successful linear MOR techniques while proper orthogonal decomposition (POD) [4,5], trajectory piece-wise linear (TPWL) [8,9], missing point estimation [10], empirical grammians [11] represent some renowned nonlinear MOR methods. However, for parameter-dependent PDEs where parameter characterize the dynamics like material properties, initial value, boundary-value, geometry features, etc, the traditional MOR schemes fail to produce faithful results. As such pMOR techniques are used to introduce the parametric dependency in the reduced models with relatively less errors [12]. A review of such methods is presented in Ref. [13] along-with some new challenges in this area. A survey on all existing pMOR techniques and their comparison is shown in Refs. [14,15]. To formulate the pMOR problem, consider a largescale parametric-dependent nonlinear, multiple-input,
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