Reliability-based aeroelastic design of composite plate wings using a stability margin

In classical lamination theory (Tsai and Hahn 1980), the constitutive equation relating applied bending moments to the curvature of a symmetrically laminated plate may be written as where the respective components of the moments and curvatures are: M = {M _x , M _y , M _{x y} } ^T and κ = {κ _x , κ _y , κ _{x y} }, and the components of laminate out-of-plane stiffness matrix D are D _{i j} (i, j = 1,2,6).

The stiffness components from (1), may be expressed as a linear combination of material invariants U _{1 − 5}, laminate thickness t, and lamination parameters ξ _{9 − 12} in accordance with (Tsai and Hahn 1980; Miki and Sugiyama 1993) The material invariants, U _{1 − 5}, are are defined as where Q _{i j} are the reduced stiffness components of an individual ply with respect to material coordinate axes, which are in turn defined as where E ₁₁, E ₂₂, G ₁₂ and ν ₁₂ are the lamina longitudinal, transverse and shear moduli, and Poisson’s ratio respectively.

By defining a non-dimensional through-thickness coordinate, u = 2z/t, the out-of-plane lamination parameters are defined as where 𝜃(u) denotes the distribution of the ply orientations throughout the laminate thickness. In practice, the laminate has a discrete set of plies with orientations \([\theta _{1},\dots ,\theta _{n}]\), as shown in Fig. 1, and (8) reduces to a through-thickness summation. The lamination parameters are defined solely as functions of the ply orientations and, as such, give the directional component of the laminate stiffness. Use of lamination parameters is beneficial as they may be uniquely defined for a sequence of ply orientations using (8), and used to represent this stacking sequence using a maximum of four parameters regardless of the total number of plies. It should, however, be noted that the restriction to out-of-plane stiffnesses relies upon an assumption that the stacking sequence is symmetric about the laminate mid-plane.

The aeroelastic stability of the simple composite plate shown in Fig. 1 is modelled using the Rayleigh Ritz method combined with a modified strip theory (Wright and Cooper 2008). In this approach, the out-of-plane displacement at any point on the plate is approximated using simple polynomial shape functions which satisfy the boundary conditions, given as where w is the out-of-plane displacement, and q _i (t)is the generalised displacement of the ith mode represented by shape function γ _i (x, y). Neglecting structural damping, and utilising the approach undertaken by Stodieck et al. (2013) and Scarth et al. (2014), the equation of motion of the plate can be expressed in the classical aeroelastic form as where A is the mass matrix, B and C are the aerodynamic damping and stiffness matrices respectively, E is the structural stiffness matrix, q is a vector of the generalised displacements \(\{q_{1}\dots ,q_{n}\}\) and ρ and V are the air density and velocity. In first-order form, (10) may be re-expressed as

Noting that system matrix Q in (11) is a function of air-speed, the eigenvalues of this matrix may be found to assess the stability of the wing at a given air-speed, as well as give the frequency and damping ratio of each mode. Instability occurs when the real part of one of the eigenvalues is positive; this instability is flutter if the imaginary part is non-zero and is divergence otherwise. The aeroelastic instability speed may be found by solving (11) at multiple air-speed increments until one of the eigenvalues becomes positive. Previous work by Stodieck et al. (2013) has validated the application of this approach to the geometry shown in Fig. 1 through comparison with a finite element model coupled with Doublet Lattice aerodynamics.

In order to illustrate the described process, example plots of the eigenvalues against air-speed are shown in Fig. 2, in which λ _j denotes an eigenvalue corresponding to the j ^thmode. Plots are shown for three example sets of lamination parameters, which are used to calculate laminate stiffness in line with (2). In each plot, the instability speed is given by the lowest velocity at which an eigenvalue crosses zero on the real axis. For example, in Fig. 2a mode 2 becomes unstable at around 120m/s. This instability is flutter as the eigenvalues are complex conjugate. In Fig. 2b, mode 2 remains stable and instability instead occurs in mode 3. Figure 2c shows divergence occurring in mode 1, as the corresponding eigenvalues are wholly real.

The optimisation and uncertainty quantification work in this paper is concerned with how variations in the composite ply orientations affect the aeroelastic stability. Such trends may be visualised in the space of lamination parameters, which are themselves functions of the ply orientations, as given by (8). Example surface plots of the instability speed are shown in Fig. 3, with respect to two lamination parameter planes; Fig. 3a shows variations in instability speed with ξ ₉ and ξ ₁₀ with ξ ₁₁ = ξ ₁₂ = 0, and Fig. 3b shows variations with respect to ξ ₁₁ and ξ ₁₂, with ξ ₉ = ξ ₁₀ = 0. Such plots are obtained by determining the instability speed using the process illustrated Fig. 2, for a large range of values of ξ _{9 − 12}. It should be noted that the lamination parameters are constrained to feasible regions (Bloomfield et al. 2009), and as such, trends are only plotted for feasible parameter values.

From Fig. 3, it can be seen that the surface defined by the instability speed is a discontinuous function of the lamination parameters, and therefore of the ply orientations. Such discontinuities have been reported upon in the literature (e.g. Haftka 1973, Housner and Stein 1974, Georghiades and Banerjee 1998, Kameyama and Fukunaga 2007), and found to be caused by a modal interchange phenomenon, in which changes in model parameters give rise to different types of aeroelastic instability, characterised by a marked change in mode-shape. In order to more clearly investigate this behaviour, an example plot of instability speed against ξ ₁₁ is shown in Fig. 4, assuming constant values for ξ ₉, ξ ₁₀ and ξ ₁₂. Such a plot is essentially a means of visualising trends with varying bend-twist coupling behaviour. Instability speeds are shown in this plot for three different types of aeroelastic instability; divergence and two types of flutter which are referred to as ‘flutter 1’ and ‘flutter 2’. The critical instability speed, given by the minimum air-speed at which instability occurs, is highlighted in this plot as a solid line.

From Fig. 4 it can be seen that multiple types of instability may be possible for a given value of ξ ₁₁; there is a region of Fig. 4 in which both flutter 1 and flutter 2 occur at different speeds, and similarly, a region in which both flutter 2 and divergence are possible. It should be emphasised that in practice the wing would be destroyed by the instability which occurs at a lower speed, however, model output may be obtained for both speeds. It may also be noted that each instability speed is only defined for distinct ranges of ξ ₁₁, as the corresponding type of instability only occurs in these regions. This phenomenon is notable in Fig. 2, in which mode 2 becomes unstable in Fig. 2a, but not in Fig. 2b.

The discontinuities highlighted in Figs. 3 and 4 can complicate reliability analysis, as commonly used techniques such as FORM, as well as surrogate modelling techniques such as Polynomial Chaos Expansion and Kriging, are based upon an assumption of smoothness. It can be difficult to apply such techniques separately to the individual instability speeds highlighted in Fig. 4, as each type of instability is limited to a region of the parameter space which is not known a priori.

In this paper, an alternative approach is presented in which surrogate models are instead fitted to the real part of each of the eigenvalues. The benefits of such an approach are demonstrated in Fig. 5, in which the real part of the eigenvalues of the first three modes are plotted against ξ ₁₁ for the same lamination parameter range used in Fig. 4, assuming an air-speed of 120m/s. The distance from instability of the eigenvalue with maximum real part, highlighted as a solid line in Fig. 5, can be thought of as a stability margin for a given air-speed. Such a margin is essentially a measure of the damping of the most critical mode. Although this margin is itself a non-smooth function of ξ ₁₁, it may be approximated by fitting multiple surrogate models to the real parts of each of the eigenvalues which, unlike the instability speeds, are defined across the full range of ξ ₁₁. A reliability-based optimisation, and surrogate modelling strategy based upon this principal is outlined in subsequent sections.

Due to the high computation time associated with reliability-based design, it can be desirable to estimate the probability of failure using surrogate models which may be evaluated in a fraction of the time required by the model itself. Fitting surrogate models to the aeroelastic instability speed can be complicated by the fact that the critical instability speed is a discontinuous function of model parameters, as discussed previously and shown in Fig. 4. In the previous section, an optimisation strategy was presented in which failure probabilities are instead calculated using a stability margin evaluated at the design air-speed. The motivation behind this method is that the stability margin may be estimated using surrogate models fitted to the individual eigenvalues, which are defined for the entire input parameter space and therefore doe not require that model inputs are partitioned. This surrogate modelling approach is discussed in detail in this section, and demonstrated in an uncertainty quantification case study.

A regression-based surrogate modelling approach for approximating scalar-valued quantities may be described as follows. The surrogate model is first trained using a small set of n samples \(\{\mathbf {x}^{(1)},\dots ,\mathbf {x}^{(n)}\}\). The bracketed superscripts are a sample index, with each x ⁽ⁱ⁾ being a d-dimensional vector of model parameters. A surrogate model, \(\hat {f}({\mathbf {x}})\), is fitted to in some way minimise the approximation error based upon model outputs at these data points. Model outputs may subsequently be approximated at ‘test’ point x as \(\hat {f}(\mathbf {x})\). This approach is common to many surrogate modelling techniques, with differences arising in the mathematical description of the surrogate model and the means by which it is fitted. In this paper, Gaussian process emulators (Rasmussen and Williams 2006; Oakley and O’Hagan 2002) are used. This section is concerned with how this approach may be modified in order to emulate multiple eigenvalues, and subsequently predict the stability margin based upon the maximum emulated value. An overview of the approach is provided in Algorithm 1, with detailed description of the various components provided in the following subsections.

In Algorithm 1, the out-of-plane lamination parameters are used as inputs to the surrogate model. Each of the x ⁽ⁱ⁾ terms described above is therefore a four-dimensional lamination parameter sample. This use of lamination parameters has been shown to be advantageous (Scarth et al. 2014), as a small number random variables may be used to represent ply orientation uncertainty, regardless of the number of plies. Lamination parameter samples are obtained in steps 2 and 7 of the algorithm through simulation of (8).

In step 5 of Algorithm 1, multiple Gaussian process emulators are fitted to the real part of each of the eigenvalues of (11). The aeroelastic model output gives rise to n samples of eigenvalues corresponding to m modes, however, there is no guarantee that modal eigenvalues will be outputted in the same order from one sample to another. Before fitting the emulators, it is therefore necessary to sort the eigenvalues such that results corresponding to the same mode are grouped together across all samples. This sorting is achieved by comparing the mode-shapes of each sample with those of a reference sample using the Modal Assurance Criterion (MAC), and grouping together results with the most similar mode-shape. In this investigation, the reference sample is arbitrarily chosen as the first sample in the data set. It should, however, be noted that in order to ensure that this sample captures all of the potential modes across the spread of behaviour, it may be necessary retain a larger number of modes than is necessary for a deterministic analysis. For two mode-shape vectors, ϕ _i and ϕ _j , the MAC is defined as (Wright and Cooper 2008)

Gaussian process emulators are a form of surrogate model which may be used to approximate continuous-valued functions. In the proposed approach, multiple Gaussian processes are used to emulate the real part of each of the eigenvalues for a given air-speed. A Gaussian process is a distribution over functions, which represents model output at fixed input parameter values as a Gaussian distribution rather than a deterministic value (Rasmussen and Williams 2006). Suppose the model output is scalar-valued and may be represented as a function, y = f(x). Suppose also that the value of this function is known for a set of n training samples \(\{\mathbf {x}^{(1)},\dots , \mathbf {x}^{(n)}\}\), with each \(\mathbf {x}^{(i)}\in \mathbb {R}^{d}\) being a sample of a d-dimensional parameter vector x. The Gaussian process is described by its mean and covariance functions, which may be parameterised as where β is a vector of regression weights, h(x)is taken as {1, x ^T}, σ ² is a scaling factor, and B is a diagonal matrix of length-scales which govern how much output f(x)varies with changes to input x. The emulator is fitted by determining the conditional distribution with respect to the training data, and finding values for the unknown hyperparameters, β, σ, and B. Maximum likelihood estimation is used to find optimal values for parameter B. The remaining hyperparameter dependencies are inferred following the Bayesian approach of Oakley and O’Hagan (2002). The resulting emulator may be simulated as a Student-t process with mean m ^∗(x), and covariance \(\hat {\sigma }^{2}c^{*}(\mathbf {x},\mathbf {x^{\prime }})\), given by where t is a covariance vector such that t _i = c(x, x ⁽ⁱ⁾), A is defined as A _{i j} = c(x ⁽ⁱ⁾, x ^(j)), and H is defined as \(\boldsymbol {H}^{\mathrm {T}} = \{\mathbf {h}^{\mathrm {T}}(\mathbf {x}^{(i)}),\dots ,\mathbf {h}^{\mathrm {T}}(\mathbf {x}^{(n)})\}\). A distinction is also made between x, which denotes a test point for which the value of f(x) is to be predicted, and x ⁽ⁱ⁾ which is a training data point for which f(x ⁽ⁱ⁾) = y _i is known. In this paper, a simplified approach is taken in which the model output is approximated using the emulator mean function, given by (20).

In this section, the surrogate modelling approach described above is demonstrated in an uncertainty quantification case study. Uncertainty is modelled as an independent and identically distributed, additive Gaussian error applied to the orientation of each ply, with zero mean and standard deviation of five degrees. Sixteen example layups have been chosen, in which all combinations of a stacking sequence parameterised as [𝜃 ₁, 𝜃 ₂, 𝜃 ₃, 𝜃 ₄,0₂,90₂] _S are modelled, with 𝜃 _{1 − 4} restricted to − 45° or 45° . In these examples, the stability margin is calculated at an air-speed of 120m/s.

The approach is validated by comparing results obtained using the surrogate model, against direct Monte Carlo Simulation using 10,000 model runs. Convergence studies have been undertaken in which the number of samples used to train the surrogate model is increased until a sufficiently good agreement is found with Monte Carlo estimates of the Probability Density Function (PDFs). Accuracy of the model is assessed using the Root Mean Square Error (RMSE) of predictions of the Monte Carlo data set, defined as where N is the total number of Monte Carlo samples, \(\hat {y_{i}}\) is the i ^th sample predicted by the surrogate model, and y is the i ^th sample of outputted by the actual model.

In practice it may not be possible to validate surrogate models using such a large data set. A more efficient alternative is to use cross-validation, whereby the training samples are partitioned into two sets, one of which is used to train the surrogate model, and a second which is used to assess the accuracy of the model (Efron and Gong 1983; Stone 1974). For example, in leave-one-out cross-validation, a single sample is omitted when fitting the surrogate model, and the difference between the model output and surrogate prediction is measured for this sample. The process is repeated for each of the training samples, and error determined using measures such as the Mean Square Error (Rasmussen and Williams 2006).

An example convergence plot of the RMSE of the surrogate model for a [45₂,− 45₂,0₂,90₂] _S laminate is shown in Fig. 6. From this plot it has been identified that 30 training samples is sufficient to achieve convergence. This procedure has been repeated for each example layup, and the resulting number of required training samples is listed in Table 2. Estimates of the mean and standard deviation obtained using both the Gaussian process emulator and direct Monte Carlo Simulation are also shown, in order to demonstrate that a good agreement is achieved. Example PDFs determined using both the surrogate model and direct Monte Carlo Simulation are shown in Figs. 7, 8 and 9. Results are shown for Gaussian processes trained with an increasing number of samples, in order to further demonstrate the convergence.

The PDF for the [45,− 45₃,0₂,90₂] _S laminate, shown in Fig. 7a, is unimodal. The equivalent PDF for the aeroelastic instability speed of this laminate is shown in Fig. 7b. It can be seen that this PDF is bimodal, which may be attributed to a mode-switch, in which the uncertainty causes the inputs to cross a discontinuity. In this case, each peak corresponds to a different type of flutter. In this case, using the stability margin therefore considerably simplifies the PDF.

It is not always possible to simplify behaviour such that the PDF is unimodal. For example, the PDF shown in Fig. 8 is bimodal due to a switch in the eigenvalue which is closest to instability, from the first to the second mode. As such, the two peaks of the stability margin PDF in Fig. 8c may be attributed the first and second modes respectively. The PDFs of the individual eigenvalue real parts, shown in Fig. 8a and b, are unimodal as the underlying function is smooth. As such, by fitting separate Gaussian Process Emulators to each of the eigenvalue real parts as described above, the stability margin may be emulated as two separate, smooth functions. From each of the plots in Fig. 8, it can be seen that using this approach a good agreement is achieved with Monte Carlo results using 30 training data points.

Figure 9 is an example of the most complicated possible behaviour, as the first mode becomes non-oscillatory, and as such, the real part of the eigenvalue for this mode is a non-smooth function of model inputs. Such behaviour may be noted in the eigenvalue plot of Mode 1 shown in Fig. 5. The PDF for this mode, shown in Fig. 9a, is therefore bimodal, with the lower peak attributable to the real part of the complex conjugate oscillatory root, and the upper peak the most-critical branch of the real-valued non-oscillatory root. The Gaussian Process Emulator does not accurately predict behaviour in the vicinity of this non-smooth point. It is, however, the upper peak of this PDF which is critical, at which point the eigenvalues are locally smooth and the emulator gives accurate predictions. The PDF of the stability margin, shown in Fig. 9c, may therefore be emulated using 30 training data points despite the inaccurate prediction of the non-smooth behaviour.

In the above examples, a good agreement with Monte Carlo Simulation is typically achieved using between 20 and 40 training data points. This observation corresponds to an order of two magnitudes reduction in the required number of model runs, compared to direct simulation.