A Non-intrusive Reduced-order Model Based On POD And Its Application

Reduced-order models(ROMs)can approximate unknown flow fields efficiently and accurately by extracting main characteristics from snapshots.As a result,they are widely used in fluid problems such as aerodynamic shape optimizations,flow controls and so on.However,traditional ROMs rely on the governing equations,which leads to the result that their application in engineering and scientific research are limited: On the one hand,IROMs are difficult to apply when the numerical source codes are unavailable;on the other hand,it is also difficult to apply IROMs to large-scale problems with high-dimensional design space.To deal with this,nonintrusive reduced-order models(NIROMs)which don't depend on governing equations have been studied and developed.Among all kinds of ROMs,the Proper Orthogonal Decomposition(POD)has been developed and applied widely,and the NIROMs based on POD are called POD-NIROMs.However,there have existed several shortcomings in POD-NIROMs' research until now.Firstly,few POD-NIROMs have been built for parameterized timedependent problems,which have high application values in practical problems.Secondly,the greedy sampling algorithm is a key point for ROMs to maintain the robustness and reduce the computational cost in the offline phase,but few greedy sampling algorithms have been presented for POD-NIROMs until now.Finally,POD-NIROMs' low approximation accuracy in strong non-linear problems has restricted their application in transonic flows.This paper aims to construct POD-NIROMs for steady and unsteady problems,present a greedy sampling algorithm to maintain the POD-NIROM' s high efficiency and robustness,couple with domain decomposition and Artificial Neural Network to improve POD-NIROMs' approximation accuracy in transonic flows,and finally apply the models to practical problems including aerodynamic shape optimizations and flow control problems.The research work of this paper is summarized as follows:a)The POD-NIROM has been improved for parameterized time-dependent problems,by increasing the computational efficiency and approximation accuracy.For parameterized time-dependent problems,the POD-NIROM has adopted a two-level POD to extract both spatial and temporal modes respectively,and used a penalized Least Square Method as well as a Radial Basis Function to approximate coefficients of POD modes.To ensure that the approximation meets boundary conditions and initial conditions,an auxiliary term has been introduced,which is solved by building another ROM.In this paper,a NIROM of POD-RBF has been adopted to decrease the computational efficiency of solving auxiliary functions.And a RBF?QR has been used to approximate coefficients of POD modes which overcomes the possible problems of the standard RBF,caused by the fact that standard RBF is very sensitive to the fitting parameters.b)An iterative greedy sampling algorithm has been proposed for POD-NIROMs to improve the robustness in the whole paramter space as well as increase the computational efficiency in the offline stage.The greedy sampling starts iteratively from initial sampling and search the point hard to predict in the design space through an error estimator.And a termination strategy has been used to determine whether the sampling terminates.In order to take both accuracy and efficiency into account,initial samples only include corner points of the parameter domain,and the termination strategy is determined by the approximated errors at the current step.An error estimator has been proposed based on the distance of POD modes' coefficients and revised by the reduced basis functions.In the diffusion problem,the error estimator can always provide the sample with the largest prediction error of the POD-NIROM at each iterative step.In the validation cases inlcuding a lid-driven cavity problem and a problem of NACA0012airfoil's flow,the greedy sampling algorithm has provided efficient samples and the PODNIROM built from the samples detemined by the greedy sampling has maintained a high approximation accuracy throughout the parameter domain.c)By coupling with a domain decomposition and an artificial neural network,a PODNIROM has been proposed for transonic flows,in order to improve the ordinary PODNIROM's approximation accuracy in nonlinear regions.Because of strong non-linear structures such as shock waves,the POD-NIROM's approximation accuacy is affected in transonic flows.To deal with this problem,the POD-NIROM has been improved in extracting reduced basis as well as approximating POD modes' coefficients.When extracting reduced basis,a Domain Decomposition has been introduced to extract the modes separately for the regions hard-to-predict,so as to provide a better reduced-order subspace.When interpolating POD modes' coefficients,an artificial neural network with multi-hidden layers has been used to improve the accuracy for non-sample points.By coupling with the domain decomposition and the ANN together,the improved POD-NIROM has presented a high approximation accuracy for the transonic flow fields of airfoils.d)The prediction of the ONERA M6 wing's aerodynamic loads and the drag reduction design of RAE2822 airfoil has been carried out to evaluate the POD-NIROM's application value in transonic flows.The POD-NIROM has predicted the aerodynamic loads of ONERA M6 wing accurately under varying aerodynamic conditions and variable geometric plane parameters.The results indicate that the POD-NIROM has presented a high ability to improve the computational efficiency of the highly reliable static aeroelastic numerical analysis method.In the drag reduction optimization of RAE2822 airfoil,an acceleration strategy has been developed firstly,which regards the POD-NIROM's approximation as the CFD solver's initial flow fields.The strategy can significantly improve the efficiency of solving the samples' flow fields.Moreover,the improvments of POD-NIROM in transonic flows can further accelerate the calculation of transonic flow fields.In the aerodynamic shape optimization design of RAE2822 airfoil,the computational efficiency of computing samples' flow fields has been improved by 41.25% with this acceleration strategy.e)Approximation of tandem cylinders' flow fields as well as a flow control problem of the GAW1 airfoil's separation have been studied,to evaluate the application value of PODNIROM in time-dependent problems.For tandem cylinders' unsteady flows,a POD-NIROM has been built with varying Distance-diameter ratio and Reynolds number.The POD-NIROM can provide accurate pressure fluctuations of the flow,And based on the POD-NIROM's prediction,the variation trend of the critical distance-diameter ratio with Reynolds number has also been concluded.In the flow control of GAW1 airfoil's separation at a high angle of attack,the effects of the blowing and suction control on the flow field and aerodynamic coefficients has been studied firstly.By comparing a uniform suction control with a dynamic blowing and suction control,it has been found that the dynamic blowing and suction can provide a higher lift coeffcient and a lower energy loss.Finally,for the sake of achieving an optimal dynamic blowing and aspiration control,a cost function has been proposed by considering the flow separation's suppression,the lift-drag characteristics and the energy loss together.And a PODNIROM has been built for unsteady vorticity field with varying control parameters of dynamic blowing and suction.Based on the POD-NIROM's approximation,the optimal parameters' values have been determined quickly.Compared to the initial dynamic blowing and suction control,the optimal one has reduced the drag coefficient by 43.4% and the energy loss by 42%.