Reduced order modeling of fluid/structure interaction.

Abstract

This report describes work performed from October 2007 through September 2009 under the Sandia Laboratory Directed Research and Development project titled "Reduced Order Modeling of Fluid/Structure Interaction." This project addresses fundamental aspects of techniques for construction of predictive Reduced Order Models (ROMs). A ROM is defined as a model, derived from a sequence of high-fidelity simulations, that preserves the essential physics and predictive capability of the original simulations but at a much lower computational cost. Techniques are developed for construction of provably stable linear Galerkin projection ROMs for compressible fluid flow, including a method for enforcing boundary conditions that preserves numerical stability. A convergence proof and error estimates are given for this class of ROM, and the method is demonstrated on a series of model problems. A reduced order method, based on the method of quadratic components, for solving the von Karman nonlinear plate equations is developed and tested. This method is applied to the problem of nonlinear limit cycle oscillations encountered when the plate interacts with an adjacent supersonic flow. A stability-preserving method for coupling the linear fluid ROM with the structural dynamics model for the elastic plate is constructed and tested. Methods for constructing efficient ROMs for nonlinear fluid equations are developed and tested on a one-dimensional convectiondiffusion-reaction equation. These methods are combined with a symmetrization approach to construct a ROM technique for application to the compressible Navier-Stokes equations.

A Stable Galerkin Reduced Order Model (Rom) for Compressible Flow

“…It turns out that the L 2 (Ω) inner product does not correspond to an energy integral for equations of compressible flow, such as the ones considered herein [1,2,3,8]. The practical implication of this fact is that if the L 2 (Ω) inner product is selected as the inner product defining the projection, the resulting ROM will not satisfy the energy conservation relation implied by the governing equations, and may exhibit an instability that is inconsistent with the physics inherent in the governing continuous PDEs [16].…”

“…Stability in particular can be an issue. A ROM constructed for the equations of compressible flow using the classical POD/Galerkin approach might be stable for a given number of modes, but unstable for other choices of basis size [16,1,2,3,8]. This leads to obvious practical limitations of the model: the ROM solution can blow up in finite time.…”

“…Mathematically, the energy is expressed as an inner product. In [1,2,3,8], it was demonstrated by the first author, Kalashnikova, et al. that the stability of the Galerkin projection step of a ROM is closely tied to the stability of the resulting model.…”

“…In the present work, the earlier analysis [1,2,3,8] is extended to the linearized compressible Navier-Stokes equations. It is demonstrated that the key ingredient to a stable Galerkin ROM for this equation set is once again a proper choice of inner product for the Galerkin projection step of the model reduction.…”

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

“…Recently, there have been some works addressing reduced order modelling of fluid and solid coupling problems [25,26,27,28,29,30]. In the work of [31], the authors presented a projection based method for contact problems.…”