Abstract

Non-Linear unsteady Aerodynamic Response Approximation using Multi-Layer Functionals

F. D. Marques

Universidade de São Paulo

Escola de Engenharia de São Carlos

Laboratório de Aeroelasticidade, Dinâmica de Vôo e Controle, LADinC

Av. Trabalhador Sancarlense, 400

13566-590 São Carlos, SP. Brasil

fmarques@sc.usp.br

J. Anderson

Department of Aerospace Engineering

University of Glasgow

G12 8QQ Glasgow

Scotland. UK

Introduction

Modelling and prediction of motion-induced unsteady aerodynamic loads in the transonic regime presents a significant challenge in aeroelastic design and analysis. The inherently non-linear relationship between the flow parameters, motion history and the aerodynamic response, which results from dynamic shock excursion and other compressibility effects, demands the use of models with a high degree of physical fidelity. Recent research effort has focused on computational procedures based on numerical solution of the governing fluid dynamic equations (Nixon, 1989). While such models are entirely appropriate as a basis for numerically computed response studies, their suitability for aeroelastic control system design and simulation is less apparent.

Empirical evidence suggests that, for a range of Mach numbers in the transonic regime, a functional description of the unsteady aerodynamic force response is justified (Tobak and Chapman, 1985). The existence of a unique non-linear aerodynamic force response functional appropriate to a particular flow regime is, necessarily, inferential. Non-uniqueness of the unsteady aerodynamic force response is generally associated with certain types of degenerate flowfield behaviour. In particular, aerodynamic hysteresis, shock waves excursion, flow instability and bifurcation have been identified as key elements in the breakdown of unique, single-valued behaviour of the aerodynamic force response. Flows admitting dynamic shock excursion offer a potentially rich source of mechanisms for the realisation of such phenomena. However, computational and experimental evidence suggests that, for certain classes of flows, unique single-valued behaviour of the aerodynamic force response is observed over a range of flow parameters and motion histories. The basic properties of the aerodynamic functional depend on the nature of the flow regime with which it is associated and the class of admissible motion histories for which it is defined.

Tobak and co-workers (Tobak and Pearson, 1964, Tobak and Schiff, 1981 and Tobak and Chapman, 1985) have developed a hierarchical class of functional aerodynamic force response models sufficiently general to encompass a broad range of flow regimes and motion histories. Although explicit representation of the aerodynamic force response functional is generally unavailable, its notional existence permits a succinct representation of the aerodynamic force response. In addition, several methods exist to identify approximate aerodynamic force response functionals from known characteristics of the motion history and aerodynamic response.

Alternative functional forms can also be used to represent non-linear unsteady aerodynamic load responses. A convenient approach is based on the Volterra functional series (Silva, 1993). Here, the aerodynamic force response functional is approximated by an infinite series of multi-dimensional convolution integrals of increasing order. However, the Volterra functional form is generally difficult to implement. Other approaches applicable to the identification of non-linear dynamic system models, for example, the Wiener methods and block-oriented methods (Billings, 1980 and Schetzen, 1980), present similar complications in the context of unsteady aerodynamic load response modelling.

Recently, an alternative approach to the approximation of non-linear functionals has been proposed by Modha and Hecht-Nielsen (1993) which provides a unified framework for the input-output description of a large class of non-linear dynamic systems. The so-called multi-layer functionals are a new parametric family of real-valued mappings defined by a non-linear combination of linear affine functionals on arbitrary normed linear spaces. Moreover, the approximate functional form is conveniently realised by a temporal neural network. The utility of artificial neural networks in non-linear system modelling is well-documented (Haykin, 1994). Identification of an appropriate neural network model is achieved via a supervised training process in which a limited sample of system input-output data sets is presented to the neural network. In general, the neural network architecture and parameters are adjusted to minimise a measure of the error between the neural network and the sample outputs. For a sufficiently broad sample of training data, the inherent generalisation properties of the neural network enable prediction of outputs in response to arbitrary inputs. An advantage of the neural network representation is that it readily accommodates multiple-input/multiple-output system descriptions. In the context of aeroelastic design, a neural network model of the unsteady aerodynamic response can be combined with a standard structural dynamic model for the purposes of control system design and/or aeroelastic simulation. Furthermore, static freestream parameters (such as, the Mach number) are readily accommodated in the neural network model thereby ensuring validity over a range of flow conditions.

In the context of unsteady aerodynamic loads response modelling, the use of multi-layer functionals has been firstly applied by Marques and Anderson (1996a, 1996b). In their work, the weakly non-linear behaviour of the unsteady normal force coefficient response on pitching 2-D airfoils in subsonic flow regime has been modelled by a temporal neural network in discrete-time, the finite impulse response (FIR) neural network. The approach has shown to furnish an efficient model form for unsteady aerodynamic load response characteristics.

This paper is concerned with the use of multi-layer functionals in the representation of motion-induced unsteady transonic aerodynamic loads response. A brief account of the approximation of non-linear functionals by temporal neural networks is presented. This is followed by a description of a neural network adaptation procedure based on a genetic algorithm and a variation of the simulated annealing algorithm in which both the neural network architecture and neural network parameters are optimised for multiple training sets. To demonstrate the feasibility of the approach, the scheme is used to identify a FIR neural network model of the motion-induced unsteady aerodynamic lift force and pitching moment response of a 2-D NACA0012 airfoil for a range of Mach numbers in the transonic regime.

Nomenclature

Multi-Layer Functional Representation of Non-Linear Dynamic Systems

Multi-layer functionals (Modha and Hecht-Nielsen, 1993) provide a convenient input-output representation for a broad range of non-linear dynamic systems and extend the concept of the universal approximation theorem (Cybenko,1989) to arbitrary input-output spaces. Multi-layer functionals comprise a class of non-linear functionals represented as a non-linear combination of linear functionals. For example, any continuous time-invariant single-input/single-output system with input u(t) and output y(t) can be approximated by a generally non-linear functional relation of the form

where z _j, q_j Î Â, L_j denote linear functionals of the input history u_t, and k is the number of process units. Here, j is a bounded, continuous function.

Typically, the linear functionals, L_j, are of the convolution type defined as

where h_j is the unit impulse response of process unit j due to u(t), for j=1,...,k.

Multi-Layer Functional Realisation via Temporal Neural Networks

Examination of Eq. (1) reveals a close correspondence with the definition of a temporal neural network (Haykin, 1994 and Wan, 1990). Temporal neural networks comprise the category of neural networks represented by a spatio-temporal neuron model joined by connecting links called synapses in which the synapses are modelled by linear, time-invariant, continuous-time filters and where each neuron modifies its inputs through an activation function.

Figure 1 illustrates the temporal neuron model in which the temporal behaviour of synapse i belonging to neuron j is described by an impulse response h_ji(t). For the input x_i(t) denoting the excitation applied to synapse i (for i=1,...,p), the synaptic response is determined by the convolution of the impulse response h_ji(t) with x_i(t). For a neuron j with a total of p synapses, the associated activation potential v_j(t), due to the combined effect of the inputs and the biases q_j, is given by,

The neuron output y_j(t) is obtained by applying the activation function, j (e.g. a sigmoidal function) on v_j(t); that is,

A multi-layer temporal neural network is formed by composing layers of neurons. A schematic representation of a single-input/single-output multi-layer network architecture, composed of neurons modelled by Eqs. (3) and (4) and distributed in two hidden layers, is illustrated in Fig. 2. Here, each of the neurons is depicted as a simplified representation of that shown in Fig. 1.

The functional approximation defined by Eqs. (1) and (2) can be interpreted as a special form of multi-layer temporal neural network possessing a single hidden layer. In practice, a more robust form of functional approximation is adopted which utilises a network architecture with multiple hidden layers. This type of network is easily generalised to accommodate multiple-inputs (including multiple static input parameters) and multiple-outputs.

FIR Neural Network Model

From a computational point of view, it is convenient to assign a finite memory, T, to the synaptic filter and to approximate the convolution integral in Eq. (3) by a convolution sum. Consequently, the continuous-time variable t is replaced by a discrete-time variable n defined by t=n Dt, where n is an integer and Dt is the sample interval. Equation (3) is then approximated as,

where t_ji is the number of delay units of the filter in synapse i belonging to the neuron j, and w_ji(l) is a weight value at time-delay l.

Equation (5) describes the expression for the activation potential of the finite impulse response (FIR) neuron model. The neural network composed of FIR neurons is referred to as a FIR neural network.

FIR Network Identification via Supervised Training

Identification of an appropriate FIR network model is assessed via a supervised training process in which a limited sample of system input-output sets is presented to the network. Where the network architecture and network delays are prescribed, the synaptic weights can be trained using a temporal version of the back-propagation algorithm (Haykin, 1994 and Wan, 1990). However, prescribed networks of this kind may exhibit poor generalisation properties. An efficient network optimisation scheme can be formulated via a genetic algorithm (Goldberg, 1989).

Network Adaptation and Genetic Algorithm

Genetic algorithms (Goldberg, 1989) are a type of evolution based search algorithm which manipulate sets of possible encoded solutions for a problem. The elements of a conventional genetic algorithm comprise:

A genetic algorithm (Goldberg, 1989) generally starts with a randomly initialised population. Each individual is evaluated by decoding its chromosome and applying the fitness function. New individuals are the result of combining individuals from the original population. A genetic algorithm is facilitated by the operations of:

The process is repeated until a new complete population is established, thereby completing a generation. The algorithm is further iterated only if a termination criteria is not satisfied.

The genetic algorithm is used as part of a supervised training process to obtain an optimal architecture for the FIR network while, simultaneously, identifying the synaptic weights. To obtain this, the algorithm interprets each FIR neural network as an individual belonging to a population. The associated chromosome is a sequence comprising the time-delays and weights per connection. The measure of the network fitness, f, is defined by the inverse of the sum of squared errors between the desired and the network outputs; that is,

where N_c is the number of input-output training sets, N_s is the total number of time samples per training set, d_k(n) is the desired output of training set k at discrete-time n and y_k(n) is the corresponding network output.

The inclusion of multiple data sets in the definition of the network fitness function differs from normal practice where a single (extended) data set is commonly adopted. The main advantage of the present approach lies in the exposure of each of the networks in the population to a broad class of input history, thereby ensuring good generalisation properties. The networks in the population are constrained to maintain certain basic features; that is, they must present a multi-layered architecture of biased neurons without missing connections between hidden layers, all hidden neurons are non-linear (sigmoid activation function), and all output neurons are linear.

The chromosome is represented by a string of constant length irrespective of the architecture encoded within it. This is attained by assuming a FIR network architecture with bounded parameters. The chromosome size depends on the limiting FIR network considered. It is a string which records the information necessary to decode any feasible network within the pre-defined bounded architecture. For each neuron of the limiting architecture, the string is the sequence of all time-delay values of the previous hidden layer to the neuron itself. The complete chromosome is the sequence described above for all neurons of the limiting architecture. Figure 3 depicts a generic representation for the chromosome encoding FIR networks, where is a flag indicating whether the neuron i exists or not, t_i,j is the time-delay in the connection between neurons j (in the previous layer) to i, and m is the number of neurons of the previous layer.

For each existing neuron, the respective weight vector (for each connection) and the bias value are recorded separately, but they must always be related to their respective connection and neuron, whatever the genetic operation.

Network Training Process

The supervised training process commences with an initial population of individuals. Each individual is created with a randomly generated architecture and receives random weight values from a uniform distribution (-1.0 to 1.0). The entire population is evaluated by a feedforward pass of each individual to produce a fitness distribution.

In the next step, parents are selected and, using the minimum, average and maximum fitness values in the distribution, the selection operator re-scales the fitnesses of the population via a linear rule and then conventional roulette wheel selection (Goldberg, 1989) is applied. Selected parents produce new individuals by the crossover operator. A crossover operator is used in which multiple crossover points may be chosen. Some care must be taken after the production of new individuals. Encoded FIR networks of different architectures may present problems during genetic operations. The gaps left inside the chromosomes by non-existent neurons or connections may lead to an inconsistent new individual after the crossover operation. The procedure is monitored to identify potential anomalies.

An attempt is made to correct any distortions of the new individuals' architecture thereby enabling chromosome strings to be re-arranged in the best way possible. If this is not feasible, then the new individual is discarded. If a new individual is accepted, there is also a possibility of that individual being mutated. The new individual's chromosome is swept gene by gene and for each gene the mutation operator changes its value with respect to a low probability value. Only time-delay values and neuron genes are mutated (with probability values P_t and P_n, respectively).

A final operation is applied to update weight and bias values of the new individuals. This operation consists of perturbing each weight and bias value by a zero mean, unit variance normally distributed random value scaled by a proportionality constant b . The new values of weight and bias are only accepted if they lead to a fitter network. This process is repeated several times before returning the modified individual to the population.

Following each generation, the new and old individuals are compared in terms of their fitness values and the best individuals retained for the next generation. To mitigate against the possibility of convergence to sub-optimal solutions, an additional mutation operator is used. This so-called forced mutation operator is invoked only after a pre-defined number of generations with the same fitness value for the best individual. For each member of the population, the operator randomly selects a gene to be modified. The gene is modified and the individual is tested to check if its new fitness is greater than the old one. If this is the case, the individual is accepted, otherwise it is only accepted with a probability of 10^-4. The routine is repeated a number of times and the final mutated individual returned to the population. Such a scheme is basically a variation of the simulated annealing algorithm.

Two criteria are used to terminate the genetic algorithm: (i) the number of generations exceeds a pre-defined limit, or (ii) the fitness of the best individual exceeds a pre-defined goal value.

FIR Network Model of Non-Linear Unsteady Transonic Aerodynamic Response

To illustrate the application of multi-layer functional representations to unsteady aerodynamic response modelling, a FIR neural network model of the unsteady aerodynamic force response to variations in angle of incidence of a 2-D airfoil operating over a range of Mach numbers in the transonic regime is identified (cf. Figure 4). Here, the non-linear functional relationship between lift force coefficient, C_L(t), pitching moment coefficient at ¼ chord length, C_m¼ (t), and incidence histories, a_t, for a NACA 0012 airfoil operating at a fixed value of Mach number (M=0.65) is assumed to be of the form,

The transonic aerodynamic database is created using a CFD code (Dubuc et al., 1997) based on the numerical solution of the non-linear Euler equations. Ideally, experimentally acquired data would furnish a much better database for the identification process. Nonetheless, the high costs involved and the lack of available data in the technical literature force the usage of alternative sources of unsteady transonic aerodynamic data.

To establish the envelope of incidence motions required to explore the complete range of continuous non-linear behaviour of the aerodynamic response, a series of numerical experiments has been carried out. These experiments are based on observations of variations of the pressure coefficient distribution around the airfoil due to shock waves excursion during the incidence motion. For a Mach number of 0.65, the maximum absolute value of the incidence angle that exhibits the aforementioned compressibility effects is in the range of 4.0º to 4.5º . Beyond these values, the data validity may be compromised.

The training process demands that a broad range of motion-induced unsteady aerodynamic responses be used during the non-linear identification of the FIR network model. This requirement is associated with the non-linear nature of the unsteady transonic aerodynamic response, and the need for a variety of characteristic motion histories within the chosen range for the network identification process. Here three characteristic motions comprising sinusoidal, ramp-up, and pulse-down input histories are considered. In each case the motion history is normalised with respect to the maximum incidence prior to training. For all training cases, the appearance and dynamic motion of shock waves responsible for non-linear behaviour of the unsteady aerodynamic response can be observed. The three characteristic motions for training have a sample interval of 0.002s, to adequate the representation of the input motion histories and output aerodynamic responses.

In order to assist convergence of the training process, the process is carried out in stages, with some training parameters altered from one stage to the next. There are no rules or laws to apply in the determination of the training parameters. For this reason a trial-and-error approach has been adopted, with initial parameters arbitrarily determined. Here, a three stage training process, totalling 250,000 generations, is carried out. For the maximum complexity FIR network architecture in the population: 2 hidden layers and 10 neurons per hidden layer are used. A maximum time-delay per connection of 6 is assumed. Table 1 presents the complete set of training parameters.

Figures 5 to ⁷ present a comparison between the aerodynamic responses obtained by the non-linear Euler CFD code and the respective FIR neural network outputs for sample training sets after completion of the identification process. The architecture and time-delay distribution of the identified FIR network model is shown in Figure 8.

The generalisation properties of the identified FIR network model are examined in Figures 9 to ¹⁴ by presenting arbitrary incidence motion histories to the identified model and comparing Euler CFD code and FIR network outputs.

Discussion

The ability of the FIR neural network to capture the essential characteristics of both linear and non-linear unsteady aerodynamic behaviour can be observed in simulations for a range of motion histories different from the ones used for training. In contrast to the time demanded by the training process to identify the model, the final network evaluations are fast enough to allow real-time predictions, justifying further applications in aeroelastic analysis and control design.

The presence of two hidden layers in the final network ensures functional complexity, as observed in Figure 8. The typical time-delay distribution within the network also provides features to the identified models that are consistent with the physical behaviour of unsteady flowfields. Despite the complexity of the searching space, the resulting network architecture is shown to be very simple. The simplicity of the network architecture, in association with good generalisation results, reinforces the satisfactory performance of the identification process.

For the identified FIR network model the non-linear behaviour of both lift force and pitch moment coefficients are adequately captured. Some discrepancies are associated with the pitch moment coefficient response; however, these are explained by the more severe non-linear characteristics of the pitch moment response. Nevertheless, these discrepancies do not spoil the prediction of the main features of the pitch moment response. For the lift force coefficient good agreement is obtained.

Generally, the predictive capabilities of the identified model of the unsteady transonic aerodynamic response are shown to be satisfactory for the majority of the test cases not contained in the training sets. This is particularly true for the lift force coefficient responses. When tested in the linear range of the unsteady aerodynamic response, the identified FIR network model has shown good predictive qualities, with an error in the overall responses not particularly large. The linear case in Figure 9 is obtained from a sinusoidal incidence motion with mean angle of attack equals to zero and 0.5º amplitude at frequency of 10Hz. The result suggests that during training the neural network model has not been presented with sufficient information to allow inferring about the linear behaviour of the system. The identified FIR network model seems to be only applicable to a limited range of motion-induced unsteady aerodynamic transonic response, in particular when flow non-linearities are present.

In the non-linear range of the unsteady aerodynamic response, the FIR network model presents good generalisation for low frequency oscillatory cases. Figure 10 presents a case where the sinusoidal incidence motion is half the frequency of the corresponding case used in the training sets. Few discrepancies can be observed in the pitch moment response of the FIR network model. Nevertheless, when a higher frequency sinusoidal motion is tested, as in the case in Figure 11, the discrepancies in the pitch moment response increase. Here, the frequency of the incidence motion is twice that of the corresponding case used in the training sets.

Figures 12 to 13 present more generalisation tests of the FIR network model of unsteady aerodynamic response in the transonic regime. The cases correspond to pulses, ramps, and oscillatory input motions in the range of non-linear behaviour of aerodynamic response, and all indicate good prediction properties of the FIR network model. The case in ^{Figure 14} shows that the model still gives extrapolation of the aerodynamic responses for an incidence angle history beyond the training limits.

Conclusions

Multi-layer functionals, realised by FIR networks, furnish a suitable representation of non-linear motion-induced unsteady aerodynamic loads in the transonic regime. The principal advantages associated with this form of functional representation are: (i) the ability to account for non-linearities and time-history dependencies encountered in unsteady transonic flow; (ii) the use of parametric multiple-input/multiple-output models for aeroelastic applications, allowing fast evaluation of the aerodynamic response; (iii) static parameters (e.g. Mach number) can be used as inputs to the network model, thereby increasing the range of flow conditions for which the model is applicable; (iv) the difficulties associated with standard non-linear system identification approaches are diminished.

The combination of genetic search and random search to identify a FIR network model is shown to overcome many of the difficulties associated with the standard temporal back-propagation algorithm, facilitating the manipulation of the network architectures. Generalisation test results show that, given only limited training set data, a FIR neural network model is capable of accurate predictions of unsteady transonic aerodynamic loads in response to arbitrary motion histories contained within the training envelope. Although the results for the moment coefficient response are relatively poor, good predictions of the lift force coefficient response are achieved. The complexity of non-linearities involved in the aerodynamic cases, added to the limitations on information contained in the training sets available for the identification process, represent important factors in the final FIR network model.

Acknowledgements

The first author (Marques, F. D.) acknowledges the financial support of the Brazilian Federal Research Council, CNPq, during the tenure of a postgraduate research studentship at the University of Glasgow, Scotland, and the financial support of FAPESP (grant 97/13323-8).

Manuscript received: February 2001, Technical Editor: Atila P. Silva Freire

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