Stability‐preserving model order reduction for time‐domain simulation of vibro‐acoustic FE models

Abstract

This work proposes novel stability-preserving model order reduction approaches for vibro-acoustic finite element models. As most research in the past for these systems has focused on noise attenuation in the frequency-domain, stability-preserving properties were of low priority. However, as the interest for timedomain auralization and (model based) active noise control increases, stability-preserving model order reduction techniques are becoming indispensable. The original finite element models for vibro-acoustic simulation are already well established but require too much computational load for these applications. This work therefore proposes two new global approaches for the generation of stable reduced-order models. Based on proven conditions for stability preservation under one-sided projection, a reformulation of the displacement-fluid velocity potential (u ) formulation is proposed. In contrast to the regular formulation, the proposed approach leads to a new asymmetric structure for the system matrices which is proven to preserve stability under one-sided projection. The second approach starts from a displacement-pressure (u p) description where the system level projection space is decoupled for the two domains, for which we also prove the preservation of stability. Two numerical validation cases are presented which demonstrate the inadequacy of straightforward model order reduction on typical vibro-acoustic models for time-domain simulation and compare the performance of the proposed approaches. Both proposed approaches effectively preserve the stability of the original system. formulation is that it results in a real dynamic stiffness matrix for undamped frequency-domain simulations and in a real eigenvalue problem for the computation of the undamped modes. It is the standard formulation used in most commercial FE tools for coupled vibro-acoustic analysis [10].One of the major drawbacks of element-based methods is their computational cost, which scales unfavourably with increasing problem size and frequency, restricting the overall practical applicability. In order to accurately describe the wave-like spatial character of the variables, the use of locally defined and low-order shape functions imposes minimum requirements on the number of elements used per wavelength and therefore limits the maximum element size for a given wavelength [11]. The highest frequency of interest determines the smallest wavelength present in the model, which in turn dictates the maximum element size. Because of the three-dimensional nature of acoustics, the number of elements in the model grows rapidly with increasing problem size or equivalently with decreasing element size (and thus also with increasing frequency). In order to alleviate the problems associated with computational costs, one can turn to model order reduction (MOR) techniques to reduce the number of DOFs for a desired accuracy [12]. For coupled vibro-acoustic problems, the advantages of MOR have already been demonstrated in frequency-domain analysis [10,13,14]. I...

3 Case studies of model order reduction for acoustics and vibrations

“…In [54] general conditions are derived that ensure to preserve the stability of a linear descriptor system in a one-sided projection. The semi-discretized system model in the time domainẋ (t) = x(t) + bu(t),…”

“…In [54] the definiteness of the global system matrices of the coupled vibro-acoustic finite element model equation (3.13) are investigated. While the coupled damping matrix C up is positive (semi-)definite, K up and M up are generally not positive definite due to the presence of K c and M c .…”

“…Note that also for uϕ and uϕ , the matrices of the equivalent descriptor system are indefinite such that stability is not preserved in a one-sided projection. It is shown in [54] that by changing the sign of the set of equations governing the acoustic degrees of freedom, resulting in the so-called modified u-ϕ, leads to global symmetric positive definite M muϕ and K muϕ and positive semi-definite C muϕ , ensuring that the conditions for stability for the equivalent descriptor formulation are fulfilled. Even though the modified u-ϕ formulation lacks symmetry as compared to the standard u-ϕ formulation, a one-sided projection-based MOR preserves stability and the resulting ROMs are well suited for time-domain simulation.…”

“…As the poles of the u-p formulation and the modified u-ϕ formulation are equal, as shown in [54], stability should also be preserved when converting a system from the u-p formulation to the u-ϕ formulation and vice versa. In order to do so, the blockpartitioned structure is required, which gets lost in the reduction process.…”

Model Order Reduced Transient Acoustic Finite Element Simulations with Impedance Boundary Conditions

“…While projection based model order reduction (MOR), guided by the system transfer function, is effective for transient problems, special care is required for construction of the reduction basis. This work builds on MOR techniques that require time-domain stability from the outset [8,9], and a strategy to accommodate frequency dependent damping while allowing iterative, numerically sound, construction of the reduction basis [10,11].…”