Time domain model reduction of linear viscoelastic finite element models R. Kuether 1 , K. Troyer 1 , M. Brake 1 1 Sandia National Laboratories Albuquerque, New Mexico, United States e-mail: [email protected] Abstract The aerospace and automotive industries often incorporate linear viscoelastic materials in mechanical systems to passively reduce vibration levels. The complicated geometries of these systems are most commonly analyzed using finite element (FE) methods. However, FE analyses that interrogate arbitrary boundary conditions may be prohibitively expensive since they typically require millions of equations to be solved simultaneously. The present work seeks to mitigate this computational burden by exploring various model reduction techniques of linear viscoelastic FE models (utilizing a Prony series) in the time domain. Specifically, a time domain solution restricts the reduction bases to be real vectors in order to produce real, reduced matrices. Several transformation bases computed from the system-level matrices will be compared using their upfront computational cost and resulting truncation error as metrics. A simple plate model will demonstrate these approaches and give insight into the accuracy and efficiency of our methods. 1 Introduction Linear viscoelastic materials are commonly used in mechanical engineering to passively reduce vibration levels of components. For example, constrained layer damping treatments add a viscoelastic core between two stiff components to mitigate vibrational energy. Core materials such as polymers and foams quickly dissipate energy and are excellent choices for achieving the desired performance. Viscoelastic materials are also used to encapsulate critical structural components (e.g., payloads, electronics, etc.) in order to isolate them from the potentially damaging operating environments. Because of the importance of these materials, there is a need to develop appropriate models that capture their physical behavior. Finite element (FE) analysis is a powerful technique to generate computationally models of structures with complicated geometries and constitutive laws describing the time-dependent material behavior. There are a number of time-dependent relaxation moduli that are used in linear viscoelastic constitutive models [1], however the Prony series is used here because it is easily implemented into computational tractable numerical integration schemes. A number of researchers have developed reduced order models (ROMs) of linear viscoelastic FE models with a variety of constitutive formulae. Bilasse et al. [2] studied four different Galerkin bases to generate the FE model based solution for the linear and nonlinear vibrations of viscoelastic sandwich beams. The authors explored the use of real eigenmodes, improved real eigenmodes, approached complex eigenmodes and exact complex eigenmodes. The results suggested that the two bases involving complex eigenmodes work quite well, and the real modes presented erroneous results but could be improved by linearizing the frequency dependent matrices about a non-zero frequency. Bilasse and Oguamanam [3] reduced the frequency domain equations of sandwich plates using real eigenmodes and exact complex eigenmode basis and found that complex modes work best for higher damping levels (albeit at a higher upfront computational cost), while real modes work sufficiently well for lower damping levels. In addition, a Craig-Bampton ROM was developed using the real modes computed from a tangent stiffness matrix, supplemented with residual vectors associated with the viscoelastic damping forces [4]. Ding et al [5] developed a free-interface substructuring approach with complex modes and a new form of the residual attachment modes. 3547
Time domain model reduction of linear viscoelastic finite element models
Jun 24, 2023
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