Abstract
Hybrid proper orthogonal decomposition (PODh) formulation is a POD-based reduced-order modeling method where the continuous equation of the physical system is projected on the POD modes obtained from a discrete model of the system. The aim of this paper is to evaluate the hybrid POD formulation and to compare it with other POD formulations on the simple case of a linear elastic rod subject to prescribed displacements in the perspective of building reduced-order models for coupled fluid–structure systems in the future. In the first part of the paper, the hybrid POD is compared to two other formulations for the response to an initial condition: an approach based on the discrete finite elements equation of the rod called the discrete POD (PODd), and an analytical approach using the exact solution of the problem and consequently called the analytical POD (PODa). This first step is useful to ensure that the PODh performs well with respect to the other formulations. The PODh is therefore used afterwards for the forced motion response where a displacement is imposed at the free end of the rod. The main contribution of this paper lies in the comparison of three techniques used to take into account the non-homogeneous Dirichlet boundary condition with the hybrid POD: the first method relies on control functions, the second on the penalty method and the third on Lagrange multipliers. Finally, the robustness of the hybrid POD is investigated on two examples involving firstly the introduction of structural damping and secondly a nonlinear force applied at the free end of the rod.
Highlights
The proper orthogonal decomposition (POD) is a powerful method providing the description of a highdimensional system by means of a small number of elements, called the POD modes (POMs)
The method is closely related to the Karhunen–Loeve decomposition originating from the probability theory [1] and to the principal component analysis commonly used in the field of statistics
A more theoretical reason proposed by Rempfer [19] is that while the local dynamics is guaranteed to improve by increasing the number of POMs, the global dynamical behavior of the system is not always preserved by the POD-Galerkin approach
Summary
The proper orthogonal decomposition (POD) is a powerful method providing the description of a highdimensional system by means of a small number of elements, called the POD modes (POMs). The method can, be formulated in a different way for such systems: the POD basis is sought as the solution of an optimization problem where the error in approximating any member of the snapshots set must be minimized. This leads to the same eigenproblem without adducing probabilistic arguments. This procedure is simple to implement and leads to a reduced-order model whose form is identical to the original discretized equation but with fewer degrees of freedom This method will be referred in the following to the discrete POD (PODd). The operators R and RÃ are based on the sðkÞ instead of the uðkÞ and the decomposition Eq (1) of the snapshots is modified in: uðkÞ 1⁄4 u þ aðjkÞjðjÞ 8k 2 1⁄21; M
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
