Abstract

Abstract

Abstract

Cost-effective parameteric surrogate models of unsteady aerodynamic loads acting on a flapping wing are highly desirable. They would enable real time aerodynamic load prediction, multiobjective optimisation and optimal control of intelligent flapping wing flight devices. In the present work, a parametric surrogate modeling framework for unsteady aerodynamic loads based on a non-intrusive reduced order modeling approach is presented. The unsteady flow past a plunging 2D flat plate is considered where the aerodynamic load time histories are obtained for different plunging frequencies and amplitudes using a potential flow solver. The parametric non-intrusive reduced order model (p-NIROM) for the obtained loads is constructed using a combination of snapshot proper orthogonal decomposition (POD) for dimensionality reduction and a fully connected feed forward neural network (FCNN) for modeling the input parametric dependency. Both, linear and non-linear FCNN based p-NIROM are explored and compared on the basis of load time history reconstruction accuracy. The non-linear FCNN regression for the p-NIROM is observed to generalise well for unseen parametric instances as compared to the linear approach when a systematic data sampling strategy is adopted.

Abstract

Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving solution-adaptive discrete operators, however, is a current challenge. Here we present an artificial neural network architecture for data-driven learning of problemand resolution-specific local discretizations of nonlinear PDEs. Our proposed method achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained network can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, addressing the problem of PDE-constrained extrapolation from data. We present examples on long-term forecasting of hard numerical problems including equation-free forecasting of the nonlinear dynamics of the forced Burgers problem on coarse spatio-temporal grids.

Abstract

Abstract

Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving such solution-adaptive discrete operators, however, is a current challenge. Here we present an artificial neural network architecture for data-driven learning of problemand resolution-specific local discretizations of nonlinear PDEs. Our proposed method achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained neural architecture model can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, hence addressing the problem of PDE-constrained extrapolation from data. We present demonstrative examples on long-term forecasting of hard numerical problems including equation-free forecasting of non-linear dynamics of forced Burgers problem on coarse spatio-temporal grids.