Real-time regional jet comprehensive aeroicing analysis via reduced order modeling

This paper presents a reduced-order modeling framework based on proper orthogonal decomposition, multidimensional interpolation, and machine learning algorithms, along with an error-driven iterative sampling method, to adaptively select an optimal set of snapshots in the context of in-flight icing certification. The methodology is applied, to the best of our knowledge for the first time, to a complete aircraft and to the entire icing certification envelope, providing invaluable additional data to those from icing tunnels or natural flight testing. This systematic methodology is applied to the shape/mass of ice and to the aerodynamics penalties in terms of lift, drag, and pitching moments. The level of accuracy achieved strongly supports the drive to incorporate more computational fluid dynamics information into in-flight icing certification and pilot training programs, leading to increased aviation safety

Nonlinear Model Reduction Based On The Finite Element Method With Interpolated Coefficients: Semilinear Parabolic Equations

For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its computational efficiency, which, however, is its the most significant advantage. Nonlinear dimensional reduction methods, such as the discrete empirical interpolation method, have been widely used to evaluate the nonlinear terms at a low cost. But when the finite element method is utilized for the spatial discretization, nonlinear snapshot generation requires inner products to be fulfilled, which costs lots of off-line time. Numerical integrations are also needed over elements sharing the selected interpolation points during the simulation, which keeps on-line time high. In this paper, we extend the finite element method with interpolated coefficients to nonlinear reduced-order models. It approximates the nonlinear function in the reduced-order model by its finite element interpolation, which makes coefficient matrices of the nonlinear terms pre-computable and, thus, leads to great savings in the computational efforts. Due to the separation of spatial and temporal variables in the finite element interpolation, the discrete empirical interpolation method can be directly applied on the nonlinear functions. Therefore, the main computational hurdles when applying the discrete empirical interpolation method in the finite element context are conquered. We also establish a rigorous asymptotic error estimation, which shows that the proposed approach achieves the same accuracy as that of the standard finite element method under certain smoothness assumptions of the nonlinear functions. Several numerical tests are presented to validate the proposed method and verify the theoretical results.Comment: 26 pages, 8 figure

Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

A new Petrov-Galerkin approach for dealing with sharp or abrupt field changes in discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modeling and stabilisation for the numerical solution of high order nonlinear PDEs. The approach is based on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution. The stabilization of the proper orthogonal decomposition (POD) model using the new Petrov-Galerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the Petrov-Galerkin POD model. Numerical results show the new nonlinear Petrov-Galerkin method is a promising approach for stablisation of reduced order modelling. Keywords: Finite Element, Petrov-Galerkin, Proper orthogonal decomposition, Reduced order modelling, Shock wave