The balanced proper orthogonal decomposition applied to a class of frequency-dependent damped structures

Abstract

A model reduction technique aimed at computing efficiently Frequency Response Functions of damped structures is presented. The frequency-dependent complex moduli are approximated by a mini-oscillators model, known as the Golla–Hughes–MacTavish (GHM) model, which permits to recast the original problem as a more familiar second-order, constant-coefficient system of equations. The matrix system, although much larger, is then treated by application of the Balanced Proper Orthogonal Decomposition (BPOD) which aims at approximating the transfer function matrix, or equivalently the admittance matrix, connecting forces and displacements at a specified set of points of the vibrating structure. All the necessary ingredients of the reduction strategy as well as its efficiency measured in terms of data reduction, accuracy and computational cost are shown. Two illustrative examples of increasing complexity involving a clamped cantilever beam and a realistic windshield are presented. It is shown that the admittance matrix can be approximated by matrices of very small size which computation can be speeded up via diagonalization. It is concluded that the application of BPOD combined with the GHM decomposition of the frequency-dependent algebraic system proves extremely efficient for the modeling of vibrating structures made of different materials, either viscoelastic or purely elastic.

Introduction

Viscoelastic materials [1] are widely used for passive damping treatments to control vibrations of mechanical structures. Sandwich structures with constrained viscoelastic materials are employed for instance in automotive, specially for windscreens. Since this component can easily radiate sounds, it can affect the acoustic comfort of the passengers inside the vehicle. Consequently, vibroacoustic simulations [2], [3] are necessary to limit this undesirable behavior. These simulations are generally based on the use of Finite Element Method (FEM) which often leads to very large algebraic systems and this renders optimizations procedures usually needed at an earlier design stage very prohibitive if not impossible.

Viscoelastic materials are known to be frequency- and temperature- dependent. The master curves for the complex-valued shear modulus can be measured thanks to a Dynamic Mechanical Thermal Analyzer (DMTA) and a parametric mathematical expression can be derived using existing rheological models such as the generalized Maxwell or Kelvin–Voigt models [4]. The frequency dependence of the FE matrix makes classical model reduction methods inoperative and more sophisticated techniques have emerged in the last decades to remedy this [5], [6]. Although this a not a place for a complete study, we can refer to a recent review by Rouleau et al. [7]. The modal-based reduction techniques discussed in the just quoted paper aims at describing the dynamical behavior of the whole structure and this is beneficial if for instance, one is interested in the identification of specific regions (a priori unknown) where the highest stresses or maximum displacements occur. There is another class of model reduction techniques which consists in approximating the input–output behavior, i.e. the transfer functions, of the original problem. These techniques which usually originate from the fields of numerical mathematics and systems and control can be extremely efficient as long as the internal behavior of the structure is of little interest [8]. Among them, the Krylov subspace based model order reduction [9] and the Balanced Truncation (BT) are probably the most famous and now widely used in a large range of applications in computational mechanics, electrical and control engineering. BT have been developed by Moore [10] in 1981 for the control theory. Moore developed it for first-order state-space models and, later, second-order versions have been implemented [11], [12], [13]. To compute the balanced realization (reduced model computing from the BT), one must compute the controllability and observability Gramians and find a transformation that makes them equal and diagonal so each state have the same degree of controllability and observability. The least controllable and observable transformed states may then be removed without altering the input–output behavior of the original system. Because BT, and also the Krylov subspace methods, have been developed for the treatment of linear or second order state-space systems, i.e. the matrix system must behave linearly or quadratically with frequency, they cannot be employed due to the complicated frequency dependence of the original FE matrix involving viscoelastic materials unless some kind of linearization process is employed.

One technique, which has been investigated in the context of FE discretization of poroelastic materials [14] consists in using a Taylor expansion of the matrix coefficients but leads to tedious and rather heavy algebraic manipulations. Other techniques which have been very popular for the treatment of viscoelastic materials are the Golla–Hughes–MacTavish GHM [15]) and the Anaelastic Displacement Field (ADF [16]) methods. The idea is to consider a generic form for the complex-valued modulus which allows to recast the original matrix system as a low order algebraic system (with real-valued coefficients) augmented with additional coordinates. This then allows classical model reduction by solving complex-valued eigenmodes and eigenvalues taking into account the damping properties of the material as shown by Friswell and Inman [17]. However, this operation can be quickly computationally expensive. For instance, Vasques et al. [18] show that the 3M ISD112 material needs three sets of parameters in the GHM model, which means that the FE algebraic system associated with the viscoelastic material must be 4 times bigger, furthermore the second-order system must be linearized first [19] in order to perform the eigendecomposition. This procedure can be avoided using BT if one is interested in computing the transfer functions between a small set of input–output variables. Friswell and Inman [17] were probably the first authors to apply it on a GHM FE matrix system by finding the balanced realization using the Cholesky factors of the Gramians. More recently, Zhang et al. [20] applied the technique to the modeling of bi-dimensional layered structures.

Controllability and observability Gramians are the unique positive solutions of the Lyapunov equations which can be computationally costly and is therefore limited to small or moderate size matrices (say up to a few thousands) [21]. In order to alleviate these limitations, low-rank factors of the Gramians can be found using algorithms based on the Alternating Direction Implicit (ADI) as shown in [22]. In 2002, Willcox and Peraire [23] proposed another strategy in order to find, at relatively small cost, a good approximate of the balanced realization. Originally developed in the field of fluid dynamics, the authors of the article used the POD (Proper Orthogonal Decomposition) method of snapshots, developed earlier by Sirovich [24] in 1991, in order to compute the low-rank factors directly from the responses of the system. The method, called Balanced POD (BPOD) was originally devised for the description of time-evolution systems though alternate representation exist in the Laplace and Fourier domain [23]. It is the purpose of this work to show the applicability of this approach for solving efficiently the dynamical response of a vibrating structure made with viscoelastic materials. One direction of particular interest to us is the dynamic substructuring for laminated structures such as windshields by computing the frequency responses, or equivalently the admittance matrix, connecting forces and displacements at a specified set of points of the structure. To the authors’ knowledge, the use of BPOD for the study of vibrational motions has not yet been explored and this paper aims to present all the necessary ingredients of the method as well as its efficiency measured in terms of data reduction, accuracy and computational cost.

The paper is organized as follows. The next section briefly reminds the GHM technique which aims at recasting the original frequency-dependent system of equations into a second order system with constant matrix coefficients. The BPOD method is then presented in Section 3 which includes a discussion on the computational aspects of the method and in particular the benefit of the GHM decomposition is highlighted in this context. In the last section, the method presented here is applied to the vibration of a clamped laminated structure of rectangular shape. Numerical results show that Transfer Function Matrices can be constructed efficiently which allows extremely fast computation of frequency response functions.