Galerkin Reduced Order Models for Compressible Flow with Structural Interaction

Barone, Matthew Franklin; Segalman, Daniel J.; Thornquist, Heidi K.; Kalashnikova, Irina

doi:10.2514/6.2008-612

Cited by 23 publications

(22 citation statements)

References 27 publications

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“…The same is not true if the Galerkin projection is performed using the L 2 (Ω) inner product. In Section 2.8 and [31], it is shown for several test cases that the symmetry inner product with appropriate boundary conditions leads to a stable ROM for the linearized compressible Euler equations, whereas the L 2 (Ω) ROM with the same boundary conditions is unstable.…”

Section: Well-posedness and Stability Of The Pod/galerkin Approach Fomentioning

confidence: 99%

“…As in Theorem 2.1, {λ i : i = 1, ..., 5} denote the five eigenvalues of the matrix A n (the diagonal entries of Λ Λ Λ n ). The terms in the integrand of the boundary integral I F j in (48) are re-cast in terms of the modal representation (31), which leads to boundary terms in the ROM 5 . In matrix form, the far-field condition can be written as…”

Section: Far-field Non-reflecting Boundary Conditionmentioning

confidence: 99%

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Reduced order modeling of fluid/structure interaction.

Barone¹,

Kalashnikova²,

Segalman³

et al. 2009

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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.

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Section: Well-posedness and Stability Of The Pod/galerkin Approach Fomentioning

confidence: 99%

Section: Far-field Non-reflecting Boundary Conditionmentioning

confidence: 99%

Reduced order modeling of fluid/structure interaction.

Barone¹,

Kalashnikova²,

Segalman³

et al. 2009

Self Cite

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show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Kalashnikova¹,

Arunajatesan²

2014

Proceedings of 10th World Congress on Computational Mechanics

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Abstract. A method for constructing stable Proper Orthogonal Decomposition (POD)/Galerkin reduced order models (ROMs) for compressible flow is described. The proposed model reduction technique differs from the approach taken in many applications in that the Galerkin projection step is applied to the continuous system of partial differential equations (PDEs), rather than a semi-discrete representation of these equations. It is demonstrated that the inner product used to define the Galerkin projection is intimately tied to the stability of the resulting model. For linearized conservation laws such as the linearized compressible Euler and linearized compressible Navier-Stokes equations, a symmetry transformation leads to a stable formulation for the inner product. Preservation of stability for the discrete implementation of the Galerkin projection is

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On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

Kalashnikova

Barone

2010

Numerical Meth Engineering

156

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SUMMARYA reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L 2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well-posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well-posed and stable far-field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty-like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in

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Galerkin Reduced Order Models for Compressible Flow with Structural Interaction

Cited by 23 publications

References 27 publications

Reduced order modeling of fluid/structure interaction.

Reduced order modeling of fluid/structure interaction.

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

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

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