The Use of Dynamic Basis Functions in Proper Orthogonal Decomposition

Abstract

Currently, the usefulness of proper orthogonal decomposition (POD) is limited to computational domains with fixed meshes and fixed boundaries. This paper presents a new POD method that enables the modeling of flow through computational domains with deforming meshes and/or moving boundaries. To achieve this goal, the solution is approximated using basis functions which, although not explicitly functions of time, depend on parameters associated with flow unsteadiness. Results are shown for transonic flow through the Tenth Standard Configuration. Comparisons are made between this method and the standard approach for on-and off-reference flow conditions. This method properly captured flow nonlinearities and shock motion for cases in which the classical POD method failed.

The exact extreme response and the confidence extreme response analysis of structures subjected to uncertain-but-bounded excitations

“…where f k, c ( t ) is the nominal value of f k ( t ); n-degree space vector ˜ f k,s ( t ) is the basis function which can be obtained via the proper orthogonal decomposition (POD) method [26,27] or other identification methods [28,29] , and ˜ f k,s ( t ) also represents a basic form of uncertain loads or uncontrolled loads such as the harmonic disturbance; p k denotes the number of uncertain loads. b k, s , s = 1, …, p k is the uncertain parameter with respect to controlled load ˜ f k,s .…”

An Efficient Proper Orthogonal Decomposition based Reduced-Order Model

“…The value of the artificial dissipation parameter νii was found using an energy-based stability analysis. Therefore, artificial dissipation was added to (16) to yield ȧ = F F F(a) + ν ν νS − τ τ τK.…”

“…Both ζ and u displayed similar results. The time coefficients were calculated in three ways: (1) the line corresponding to "FOM" solved (6) for the "exact" time coefficients, (2) the line corresponding to "ROM with no BC" solved (15) for the time coefficients, and (3) the line corresponding to "ROM with BC" solved (16) for the time coefficients. Both ROM solutions were extrapolated past the sampling time of the FOM.…”

“…While this is not a comprehensive list, it is clear that these model reduction procedures have become an important part of CFD model development. Furthermore, manipulations to the traditional POD method have been performed involving modifying the form of the governing equations, e.g., through linearizing the system of equations, changing the inner product of the Galerkin projection procedure [15], or extending the POD approximation to deforming computational domains [16].…”