Maciej Balajewicz

Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation

MACIEJ J. BALAJEWICZ1†, EARL H. DOWELL2 AND BERND R. NOACK3 Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 POITIERS Cedex, France

Reduced-Order Modeling of Flutter and Limit-Cycle Oscillations Using the Sparse Volterra Series

For the past two decades, the Volterra series reduced-order modeling approach has been successfully used for the purpose of flutter prediction, aeroelastic control design, and aeroelastic design optimization. The approach has been less successful, however, when applied to other important aeroelastic phenomena, such as aerodynamically induced limit-cycle oscillations. Similar to the Taylor series, the Volterra series is a polynomial-based approach capable of progressively approximating nonlinear behavior using quadratic, cubic, and higher-order functional expansions. Unlike the Taylor series, however, kernels of the Volterra series are multidimensional convolution integrals that are computationally expensive to identify. Thus, even though it is well known that aerodynamic nonlinearities are poorly approximated by quadratic Volterra series models, cubic and higher-order Volterra series truncations cannot be identified because their identification costs are too high. In this paper, a novel, sparse representation of the Volterra series is explored for which the identification costs are significantly lower than the identification costs of the full Volterra series. It is demonstrated that sparse Volterra reduced-order models are capable of efficiently modeling aerodynamically induced limit-cycle oscillations of the prototypical NACA 0012 benchmark model. These results demonstrate for the first time that Volterra series models are capable of modeling aerodynamically induced limitcycle oscillations.

Application of Multi-Input Volterra Theory to Nonlinear Multi-Degree-of-Freedom Aerodynamic Systems

This paper presents a reduced-order-modeling approach for nonlinear, multi-degree-of-freedom aerodynamic systems using multi-input Volterra theory. The method is applied to a two-dimensional, 2 degree-of-freedom transonic airfoil undergoing simultaneous forced pitch and heave harmonic oscillations. The so-called Volterra cross kernels are identified and shown to successfully model the aerodynamic nonlinearities associated with the simultaneous pitch and heave motions. The improvements in accuracy over previous approaches that effectively ignored the cross kernels by using superposition are demonstrated.

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier-Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier-Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

Reduced Order Modeling of Nonlinear Transonic Aerodynamics Using a Pruned Volterra Series

The following paper presents a reduced-order-modeling approach for nonlinear aerodynamic systems utilizing a pruned Volterra series. The method is applied to a two-dimensional transonic airfoil undergoing forced pitch oscillations. Pruned Volterra series reduced-order-models up to fourth-order are identified and compared against computational fluid dynamics models. Very favorable accuracies are attained over a wide range of Mach number, reduced frequency and oscillation amplitude. The computational resources associated with the pruned Volterra series are demonstrated to be several ordersof-magnitude lower compared to the standard Volterra series.

Reduced Order Modeling of Transonic Flutter and Limit Cycle Oscillations Using the Pruned Volterra Series

This paper presents a reduced-order-modeling approach for nonlinear aeroelastic systems utilizing the pruned Volterra series. This approach is used to approximate the flutter boundary and LCO amplitudes of the NACA 0012 benchmark model. Pruned Volterra series reduced-order-models up to third-order are identified and compared against experimental and harmonic balance results. Very favorable accuracies are attained for the specific transonic Mach numbers tested. The computational savings associated with the pruned Volterra series are demonstrated to be several orders-of-magnitude.

Reduced order models for pricing American options under stochastic volatility and Jump-diffusion models

Abstract American options can be priced by solving linear complementary problems (LCPs) with parabolic partial(-integro) differential operators under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. These operators are discretized using finite difference methods leading to a so-called full order model (FOM). Here reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD) and non negative matrix factorization (NNMF) in order to make pricing much faster within a given model parameter variation range. The numerical experiments demonstrate orders of magnitude faster pricing with ROMs.

Reduction of nonlinear embedded boundary models for problems with evolving interfaces

Embedded boundary methods alleviate many computational challenges, including those associated with meshing complex geometries and solving problems with evolving domains and interfaces. Developing model reduction methods for computational frameworks based on such methods seems however to be challenging. Indeed, most popular model reduction techniques are projection-based, and rely on basis functions obtained from the compression of simulation snapshots. In a traditional interface-fitted computational framework, the computation of such basis functions is straightforward, primarily because the computational domain does not contain in this case a fictitious region. This is not the case however for an embedded computational framework because the computational domain typically contains in this case both real and ghost regions whose definitions complicate the collection and compression of simulation snapshots. The problem is exacerbated when the interface separating both regions evolves in time. This paper addresses this issue by formulating the snapshot compression problem as a weighted low-rank approximation problem where the binary weighting identifies the evolving component of the individual simulation snapshots. The proposed approach is application independent and therefore comprehensive. It is successfully demonstrated for the model reduction of several two-dimensional, vortex-dominated, fluid-structure interaction problems.

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range.