Efficient Surrogate-based Robust Shape Optimization ...

Efficient Surrogate-based Robust Shape Optimization for Vane Clusters Ilya Arsenyev∗ and Fabian Duddeck∗

Andreas Fischersworring-Bunk†

Technische Universit¨ at M¨ unchen, Germany

MTU Aero Engines AG, M¨ unchen, Germany.

The paper is aimed to describe the robust optimization process for a shape optimization of a low pressure turbine (LPT) guide vane cluster. The multidisciplinary analysis, needed to evaluate multidisciplinary constraints, which drive the vane cluster design, is described. Attention is paid to the different sources of uncertainty, such as manufacturing tolerances, material parameters and performance parameters. An overview of the recent methods for the Reliability-based Robust Design Optimization (RBRDO) is presented, with the focus on efficient adaptive surrogate-based method. Finally, a new surrogate-based approach is introduced, which is inspired by the Efficient Global Optimization (EGO) method1 and inverse reliability analysis.

I.

Introduction

Increasing efficiency and reliability, while reducing development and manufacturing costs are the goals for the design of advanced civil aircraft engines. To achieve these requirements, each particular component of the engine needs to be optimized. Various sources of uncertainty are often neglected when performing deterministic optimizations, which usually leads to optimal solutions with active constraints. That means, that slight deviation of the design parameters or loads from the nominal values can lead to the constraints violations, and probably cause component malfunction. Thus, especially for applications with high robustness and reliability requirements as encountered in jet design, these parameter uncertainties should be taken into account during the optimization. From the deterministic point of view, two types of parameters control the vane design: design variables and parameters, which are fixed during optimization. From the robustness point of view, all parameters can be seen as stochastic (uncertain) or deterministic. Combining these two classifications, four different parameter types should be considered: deterministic design variables, deterministic parameters, stochastic design variables and stochastic parameters. The stochastic design variables are, for example, shape parameters which include manufacturing tolerances, and can be treated as two separate parameters: a deterministic (mean) as design variable and the deviation from that value as stochastic parameter. All these parameter types are considered in the vane cluster design. The guide vane is a crucial component for the LPT design, responsible for optimal direction of the gas flow to act on the rotor blades. The design of the vane cluster is driven by multiple disciplines, such as aerodynamics, thermal-mechanics and structural mechanics. Multidisciplinary analysis (MDA) and optimization (MDO) is aimed to satisfy conflicting multidisciplinary constraints and improve the overall performance of the guide vane. As was mentioned before, stochastic treatment of design parameters needs to be introduced for obtaining robust optimal solution. In order to get accurate estimation of the design-driving quantities, costly high-fidelity simulations need to be used. The high computational costs, required for MDA simulations, as well as high dimensionality of the problem, makes direct application of the global optimization combined with sampled-based uncertainty analysis techniques inefficient. To overcome these issues, a global surrogate-based approach is proposed in this research, based on the EGO1 method and inverse reliability analysis. ∗ Research † TES,

associate, Chair of Compuational Mechanics, Arcisstr. 21, 80333, M¨ unchen. Dachauerstr. 665, 80995, M¨ unchen

1 of 16 American Institute of Aeronautics and Astronautics

II.

Vane Cluster Multidisciplinary Analysis

The design of the vane clusters is an iterative complex analysis process aimed to satisfy all the designdriving constraints, related to the different disciplines. The automatic MDA process chain, implemented for the vane cluster analysis, is shown in Fig. 1. This research is focused on the robust shape optimization of an uncooled guide vane cluster of the 2nd stage of a LPT turbine, consisting of three vanes.

Figure 1: Multidisciplinary analysis chain.

Deterministic analysis process chain The deterministic multidisciplinary analysis chain starts with the 3D aerodynamic analysis of the 2nd stage of LPT turbine stage (including vane and rotor blade) for the cruise conditions (aero-design point, ADP) and is performed using RANS TRACE solver, developed by DLR.2 Transient thermal (TT) analysis is important for accurate prediction of the life of the vane cluster, which is affected by high temperature gradients, which occur during the engine mission and lead to high thermal induced stresses in the vanes. Thermal boundary conditions are interpolated in time between each two working points of the mission, for which the boundary conditions are known from the 2D aerodynamic analysis (blade-to-blade and hub-to-tip analysis, using special time-dependent η-curves. For the current research, a simplified engine mission is used, consisting of two parts: acceleration from idle (IDLE) to takeoff (TO) conditions followed by deceleration from TO to IDLE and skipping the cruise point. Additional room temperature point (293 ◦ K) is added to the end of the mission. As can be seen in Fig. 2, where the evolution of 3D temperature fields is shown, the vane edges are much hotter (during acceleration IDLE-TO) compared to mid-camber vane regions, because thinner edges are heated up much faster than thicker central parts of the vanes.

Figure 2: Evolution of 3D temperature field.


The following structural analysis includes static, dynamic (modal) and a thermo-mechanical fatigue (TMF) analysis. The same FE mesh is used, as for TT analysis, which allows direct node-to-node temperature mapping between TT and structural models. TO conditions are used for the static analysis with the corresponding gas loads and 3D temperature field (important for temperature-dependant material properties). TMF analysis is used to compute thermal-induced stresses in the vanes during the mission, resulting from temperature gradients obtained from TT analysis and material thermal-expansion properties. These stresses are combined then with the static stresses for time discretization point. Static stresses, induced by the gas loads, are computed using single linear FE analysis, and scaling with the rpm η-curve is applied to get static stresses for every time discretization point without additional simulations. The combined stress is used to evaluate the damage accumulated at each point of the vane cluster after a certain number of engine missions. Finally, dynamic analysis is used to ensure, that no resonance will occur in the working rpm range. For this particular study, simplified analysis is performed without the casing, when the vane cluster outer hooks are fixed instead. Dynamic analysis is carried out for the TO conditions in this study. Vane cluster design parameters As was mentioned in the introduction, several different types of parameters should be considered for the robust design optimization. First of all, these are geometry parameters, controlling the shape of the vane cluster. The vane model is decomposed into an assembly of an airfoil part and vane shrouds part, which are both parameterized explicitly. The airfoil geometry is modelled using pdesk software and based on a sectionwise profile definition and subsequent section stacking to produce the three-dimensional (3D) geometry. Axial and circumferential shifting parameters sa and sθ control positions of the 2D profile sections. For the parameter interpolation over the radial position 3rd order splines are used here. Finally, the airfoil is cut to fit into the annulus collector and saved as 3D vane model. This geometry is used directly to create a CFD model. The same single vane 3D model is used to update the solid model assembly. The assembly part with the shroud’s geometry contains also parameterization of the outer shroud shape. Example of the solid vane cluster model is presented in Fig. 3(b). In total 64 parameters can be used to control the vane cluster shape.

(a) Vane parameterization.

(b) Vane cluster solid model.

As manufacturing of the vane cluster always involves dealing with manufacturing tolerances, most of the shape design parameters should considered as uncertain design variables. For example, leading and trailing edge thickness are the design variables, while corresponding manufacturing tolerances are the additive uncertain variables. Major design parameters, e.g. stagger angle are considered as deterministic design variables. Another source of uncertainty is the variation of the loading conditions. For the vane cluster analysis, design driving thermal and structural (gas) loads are defined by engine performance parameters, such as gas temperatures and pressures, measured at certain locations for different engine working points. Variation of these parameters leads to the variation of thermal and structural loads. In this research, performance temperatures variations are considered, because thermal loads are crucial for the vane cluster fatigue life. Computational costs As can be concluded from previous paragraphs, a complex and expensive simulation process is required for obtaining design-driving quantities for the vane cluster. High quality CFD meshes (approx. 400,000


cells) and structural meshes (approx. 700,000 elements) lead to the total computation time of 2-3 hours for evaluating a single design. Process parallelization increases efficiency significantly, ending with 8-12 design evaluations in 4 hours. Considering the high-dimensional parameter space and high computational costs, direct application of optimization techniques combined with uncertainty quantification (robustness/reliability assessment) is to demanding, hence surrogate modeling is used.

III.

Reliability-based Robust Design Optimization

The goal of the robust design is to find a design which is relatively insensitive to the variation of uncertain parameters. Usually not only the objective function, but also constraints drive the product/system design and it is also important to account for their variability under the presence of uncertainty. The system is reliable to a certain degree, if all constraints are satisfied with the given probabilities subject to the present uncertainties. Estimating and controlling failure probability Pf is the goal for the Reliability-Based Design (RBD). To obtain good optimal designs both of these robustness and reliability requirements should be fulfilled, which leads to the Reliability-based Robust Design Optimization (RBRDO). Robustness and Reliability measures Let’s consider a performance function f , which depends on design variables (deterministic) xd , uncertain variables (e.g. environment parameters) xu and uncertain design variables (design variables which exhibit some perturbations, e.g. shape parameters with tolerances) xud . The latter can be viewed as a combination of deterministic design parameters µud and uncertain variables δud . In general, δud can depend on µud , e.g. manufacturing tolerances can depend on the product dimensions. With this decomposition, we can denote {xd , µud } as a design vector x, and uncertain parameters vector {xu , δud } as θ. Several robustness measures were proposed in the literature within the probabilistic approach to quantify robustness, from which the mean (first moment) and the variance are the most popular. It often happens, that there is a trade-off between the mean and the variance, and simultaneous minimization of both measures leads to a multi-objective problem. One way to solve this problem is to translate it into a single-objective by using the weighted sum approach: minimize F (x) = E[f (x, θ)|x] + wV ar[(f (x, θ)|x],

(1)

where w is a specified weighting factor. Minimization of F should lead to both minimization of E[f |x] and V ar[f |x]. If the maximum allowed variance of f can be explicitly specified, multi-objective unconstrained problem can be converted into the single objective problem with the maximum allowed variance constraint. Another option is a multi-objective optimization, resulting in a Pareto front (e.g using genetic algorithms). In case of a constrained problem, let g(x, θ) be a vector of constraint functions with the deterministic constraint g(x, θnominal ) ≥ 0. As θ are the random values, gi is also a random variable with its distribution. The straightforward probabilistic reformulation of the constraint is to use the probability of failure as the measure of the system reliability: P [gi (x, θ) ≤ 0|x] ≤ Pif , where Pif is the failure probability threshold for the ith constraint. Instead of directly using probabilities as constraints, the reliability index β can be used (reliability is defined as 1 − Pf ): βi = Φ−1 (1 − P [gi (x, θ) ≤ 0|x]) ≥ Φ−1 1 − Pif = βif , (2) where βif is the minimum required reliability index for the ith constraint and Φ is the standard normal CDF. Another approach is to use the inverse reliability approach, leading to the notion of probabilistic performance measure. This is the level of the constraint, which exactly satisfies the required reliability index value: gip : P [gi (x, θ) ≤ gip |x] = Φ −βif (3) Reliability analysis Simulation-based methods A widely used class of non-intrusive uncertainty propagation techniques are the simulation-based methods. These methods can be applied to almost any uncertainty propagation problem, and given a sufficient number of system evaluations can provide the required level of accuracy for statistical moments or distributions. 4 of 16 American Institute of Aeronautics and Astronautics

The most well-known simulation method is the Monte-Carlo (MC) method. The integral values (e.g. statistical moments, failure probabilities, etc.) are estimated using large number of samples, derived from the joint PDF, and the following approximation of the integral as a sum. The MC method has low convergence rate (∼ n−1/2 , n is the number of samples), but at the same time independent from the problem dimension. This method is very demanding when applied for the estimation of rare event probabilities. Advanced sampling techniques, such as low-discrepancy sequences or Latin-Hypercube sampling, allow to improve the convergence rate. Another approach aimed at the convergence rate improvement is Importance Sampling (IS) proposed by Marshall.3 This method estimates the integral value by sampling with the modified distribution in order to increase the sampling density in the regions higher contribution to the total integral value. Based on this approach, several simulation-based methods were proposed specifically to estimate the failure probabilities, e.g. Directional Sampling, Subset Sampling (SS),4 Line Sampling. Simulation-based techniques can be used for estimating both robustness (e.g. mean, variance) measures as well as reliability measures of the system. These methods are non-intrusive and can be applied to virtually any class of uncertainty quantification problems. The drawback of these techniques is slow convergence rates, when even the most advanced methods being very computationally demanding. MPP methods The most probable failure point (MPFP) is defined as the closest failure point to the origin of the standard uncertain space. From definition if follows, that this is the point on the limit-state contour with the highest value of the joint uncertain variables PDF. Using the linearisation of the limit-state function in vicinity of the MPP, in case if it is unique, the approximation of the failure probability can be obtained (referred to as First Order Reliability Method, FORM). The Hasofer-Lind reliability index βHL is defined as signed distance between the MPP and the origin of the standard normal uncertain space: βHL = −u∗ · ∇g/ k ∇g k= u∗ · n,

(4)

where u∗ is MPFP and n is the direction, opposite to the g gradient. Failure probability is than approximated as Pf = Φ−1 (−βHL ). The main limitation of MPP-based approach is its inaccuracy in case of multiple MPPs. The methods which operate with the MPP-based approximation of the reliability index are called Reliability Index Analysis (RIA), while the ones, which compute MPP-based probabilistic performance measures are noted as Performance Measure Analysis (PMA). Both approaches are widely used within the RBDO for the efficient approximation of the failure probabilities. MPP-based reliability analysis methods require a transformation of the original uncertainty variables θ into the standard independent normal space u, which can be done using isoprobabilistic transformations, e.g. Rosenblatt5 or Nataf 6 transformation. For the RIA, in order to find βHL , the following minimization problem should be solved: minimize k u k subject to: gi = 0 (5) The PMA formulation is inverse to the RIA leading to the following minimization problem: minimize gi (u) subject to: k u k= βif

(6)

The solution u∗ of the PMA problem is called minimum performance target point (MPTP). In general, both MPP-based RIA and PMA can be solve using standard constrained optimizers, but more efficient methods exist for both approaches. For the RIA, the improved Hasofer-Lind-Rackwitz-Fiessler algorithm (iHLRF)7 for the MPFP can be used. The PMA consists in search for the minimum limit-state function value on the hyper-sphere with radius equal to the target reliability index βif . Specific algorithms developed such as the Advanced Mean Value (AMV), the Conjugate Mean Value (CMV),8 the Hybrid Mean Value (HMV)8 and Most Probable Point of Inverse Reliability (MPPIR)9 methods. The enhanced versions of HMV10 and enhanced MPPIR11 were proposed to improve the convergence for highly non-linear limit-state functions. All these methods internally handle the the target reliability sphere constraint. PMA minimizes a complicated function with a simple constraint, while for RIA a simple function is minimized subject to a complicated constraint, which is less computationally efficient.8, 12 PMA better converges for different types of probabilistic distributions, because non-linearity of the constraint is less sensitive to distribution type. MPP-based methods require much less computational effort, compared to simulations-based techniques. What is even more important in the context of RBRDO, that unlike simulation-based methods, two runs 5 of 16 American Institute of Aeronautics and Astronautics

of an MPP-based method will converge to the same solution under the same initial conditions. This can improve the convergence of the RBRDO and also simplifies the use of surrogate modeling. Reliability-based Robust Design Optimization Problem Formulation A general constrained deterministic optimization problem can be formulated as follows: minimize f (x) subject to g(x) ≤ 0, xL ≤ x ≤ xU ,

(7)

Under the presence of uncertainty θ, both objective and constraints in general can’t de considered as deterministic, and thus the optimization problem should be reformulated in the probabilistic way, using the aforementioned robustness and reliability measures: minimize F (x) = E[f (x, θ)] + wV ar[(f (x, θ)] subject to P [g(x, θ) ≤ 0] ≤ P0 , xL ≤ x ≤ xU ,

(8)

Overview and comparison of the recent RBDO methods can be found in.13 In general, RBDO methods can be divided into two-level or nested (e.g. RIA or PMA RBDO), mono-level (e.g. SLA, SLSV) and decoupled approaches (e.g. SORA, SAP). While mono-level and decoupled approaches are more computationally efficient, they can experience difficulties dealing with highly non-linear problems and have convergence and robustness issues. Results from13 shows that the nested PMA approach is the most robust and accurate. In order to solve RBRDO, the RDO and RBDO techniques should be combined. One option is to use nested RBDO approach and add robustness analysis to the inner loop. For example, nested approach with Monte-Carlo simulations to evaluate all robustness/reliability measures and evolutionary strategy for the optimization is used in,14 which is a sort of brute-force solution. More efficient combination of SLSV mono-level RBDO method with robustness objective function15 or decoupled approaches16 can be used.

IV.

Efficient Surrogate-based RBRDO approach

The high computational costs, required for MDA simulations, as well as the high dimensionality of the problem, make the direct application of the RBRDO inefficient. To reduce the number of MDA simulations and computational time, a surrogate-based robust optimization approach is proposed in this research, based on Gaussian-Process (GP) surrogates.17 In the following paragraphs, the overview of existing methods, which utilize GP surrogates for the RBRDO will be given, as well as description of the proposed method, which combines Efficient Global Optimization method and robustness/reliability assessment. Efficient Global Optimization method The proposed here method is based on the Efficient Global Optimization (EGO) method.1 The EGO method exploits the Gaussian Process (GP) surrogate models, which use the posterior (Gaussian) distribution to predict values at unsampled location. The posterior mean can be used as a surrogate for the original function, while the posterior variance gives an estimate of the model uncertainty. Using these quantities, the efficient surrogate adaptivity criterion, the Expected Improvement (EI), was proposed in.1 Let fmin = min(f1 , f2 , ..., fn ) denote the objective function value of the best point in the current training dataset. Then, the improvement we are likely to make at a new point, say x, is given by I(x) = max(fmin − F (x), 0). Using the fact that F (x) is Gaussian Process model, the expected improvement can be written in closed form as: ! ! ˆ(x) ˆ(x) f − f f − f min min + σ(x)φ , (9) EI(x) = (fmin − fˆ(x))Φ σ(x) σ(x) where Φ and φ denote the standard normal CDF and PDF, respectively. The fˆ and σ are predicted mean and variance of the Gaussian process surrogate model. The EI expression (9) includes the term proportional to (fmin − fˆ(x)) describing the possible improvement of the objective, which is responsible for a local exploitation, and the second term proportional to σ describing the predicted uncertainty of the surrogate model, which is responsible for exploration of the design space. Extension of EI to the constrained case was proposed in1 with the use of the feasibility probability. The same implementation of the EGO method, as was used previously for the vane cluster multi-disciplinary optimization18 is used in the current research. 6 of 16 American Institute of Aeronautics and Astronautics

Existing adaptive GP-based RBRDO approaches Several surrogate-based methods for efficient solution of the RBRDO problem with the use of GP surrogates were recently proposed, e.g. RDO method by Jurecka.19 The surrogate model covers the combined (design + uncertain) space and a two stage criterion is proposed to adaptively refine the surrogate: the EI criterion is used to find the infill point x∗d in the design space with sampling in the uncertain subspace to obtain robustness measures. For the found design point x∗d , the remaining uncertain variables values x∗u are found to maximize the predicted GP model variance in the uncertain subspace for improving the surrogate accuracy. Unfortunately no hints were given for the case of uncertain design variables xud . Lee and Jung proposed20 to use Constrained Boundary Sampling (CBS) for the GP surrogates refinement along the limit-state contour, based on predicted posterior mean and variance of GP. Unlike EI, the surrogate is being refined on the specified contour line, e.g. g = 0. This improves the accuracy of the surrogate for reliability analysis (RIA or simulations-based). The authors use AMV method to compute reliability indices during RBDO. Bichon21 proposed the new infill criterion, the Expected Feasibility (EF), which similarly to CBS adaptively refines the GP surrogate along the specified contour. The EF criterion is an expectancy measure that the point x belonging to the specified contour g = z: Z z+ EF (G(x)) = ( − |G − z|)P DFG dG, (10) z−

where G is a realization of GP predictor of function g and defines the region with values close to the threshold z. It is recommended to choose ∼ σG e.g. = 2σ. After surrogates are accurate enough, the Multimodal Adaptive Importance Sampling (MAIS) is being used for reliability analysis. This procedure is called Efficient Global Reliability Analysis (EGRA). The drawback of the proposed method and the method from Lee and Jung is that surrogate models are being refined uniformly over the limit-state contour lines, thus sampling a lot of points also in the regions where the failure is very unlikely to happen and which have small impact on the total probability of failure. Another approach is proposed by Dubourg,22 where GP surrogates and Subset Sampling (SS) sampling were combined into an efficient adaptive RBDO procedure. The SS is used for efficient refinement of the surrogates, where the points are added adaptively in the regions along the contour, where the failure is more likely to occur (unlike the uniform limit-state contour refinement). Pollak-He algorithm is applied together with SS to solve for RDO problem using GP surrogates. This steps are repeated until the convergence criteria is met. The stopping criterion is based on the predicted confidence bounds of failure probability, which depend on the GP surrogates accuracy.22 The strategy still relies on the gradient-based optimization which tends to search for a local optimum and can potentially lead to inaccurate results in case of poor objective function surrogate. Proposed method The main idea of the proposed strategy is to exploit the efficiency of EGO for finding the global optimal solution in the design space and combine this method with both the reliability and robustness analyses. The RBRDO surrogate-based approach is proposed, which employs the EGO method as the top-level optimizer, while PMA analysis and sampling-based robustness analysis are used for reliability/robustness analysis. The main requirement for the method, considering costly MDA simulations is to minimize the number of simulations, while providing sufficient accuracy of the results. To achieve these goals, adaptive refinement of the surrogates is required, similar to the methods described before. The main steps of the procedure are: 1. Generate a samples in the combined design-uncertain space. 2. Fit GP surrogate models for the objective and constraints using the generated samples 3. For each training point within design space perform PMA analysis 4. For each training point within design space compute robustness objective using sampling 5. Using results from steps 2 and 3, fit second level GP surrogates in the design space 6. Run step of EGO method on the second level surrogates and obtain infill points in design space 7 of 16 American Institute of Aeronautics and Astronautics

7. Find infill points in the uncertain space: • For the objective surrogate refinement, the approach by Jurecka19 is used. • For constraint surrogates, the PMA analysis is performed for infill points, and if the predicted probability of constraint being active is higher than threshold, the obtained point is added to the constraint’s training set. 8. Add infill points to all surrogates 9. Check convergence. If not converged, go to step 2 The flow of this process is depicted in Fig. 3. It is important to generate good initial sampling for fitting the initial surrogates. In case of no prior knowledge of underlying relations, uniform space-filling sampling in general will lead to better surrogate quality. The Latin-Hypercube sampling, proposed b McKay23 is used, which is being optimized for better uniformity using columnwise-pairwise updates24 combined with Simulated Annealing, similar to.25 The number of the samples is taken similar to that of the EGO method: Jones et al.26 suggest around 10 times the problem dimension based on the their own experience. According to Sobester,27 an initial sample size around 35% of the available computational budget is a safe choice and the method loses efficiency if initial sample exceeds 60% of the total budget.

Figure 3: Flow chart of the proposed methodology. Using the generated samples, GP surrogates are fitted over design and uncertain variables with the Maximum Likelihood Estimation (MLE)17 for hyperparameters. Based on these surrogates, robustness/reliability analysis is performed for each training point. With the presence of uncertain design variables, only points which lie within design bounds are selected. In order to compute mean and variance of the objective, sampling in the uncertain space is used. For the constraints, PMA analysis is performed using enhanced MPPIR11 method. This provides the value of the constraint at the most probable failure point, corresponding to the specified probability of failure threshold, which should lie within the deterministic bounds of the constraint. The second layer GP surrogates are fitted over the design space using robust objective values and constraints’ levels from PMA, which are now the deterministic constraints for the top-level optimizer. These surrogates share the points with the first layer surrogates in the design space, which enforces simultaneous refinement of the GP approximations in the design space for both layers. This can be very beneficial when a large number of uncertain-design variables are present (e.g. shape parameters with tolerances), because often surrogates get accurate enough in the vicinity of the optimum (in the design space) from the EGO


refinement. The EGO method step is performed on the second level surrogates, searching for the promising points to minimize robust objective, while satisfying reliability constraints. For each of the design infill points, found by the EGO step, the values of the uncertain variables are determined to refine the surrogates in the uncertain subspace. In case of robust objective, the point with maximize the posterior objective GP variance is found, similar to.19 For the constraints, the PMA analysis is performed for each infill points, and if the probability of the constraint being active is higher than the certain threshold (obtained PMA level is close to the constraint bound or GP surrogate has high predicted variance), than this MPP coordinates are used for the refinement. This approach allows to refine the surrogates in the uncertain subspace in the regions close to the PMA predicted MPPs, which is less expensive, than contour refinements. When all infill points for all GP surrogates are found and evaluated, the surrogates are refitted and the next iteration of the method is performed. If highest EI of infill points for last 2 iterations is less than a threshold, the iterations are stopped. The method is aimed mainly at the shape optimization problems, where the tolerances must be taken into account in order to obtain robust/reliable optimal design. In this case, the proposed formulation doesn’t increase the dimension of the problem, as design uncertain variables are not decoupled for the GP fit. Often, the refinement of the surrogate, based on the EI criterion is sufficiently accurate also for the robustness/reliability analysis without the need of additional refinement in the uncertain space, as surrogates automatically get more accurate near the optimal solution.

V.

Testing of the Method

Short column Several well-known test problems, taken from the literature, are used to evaluate the performance of the proposed method. The short column example is used often for testing various RBDO methods, and this particular formulation is taken from,21 which allows direct results comparison. The initial 21 samples are generated using optimized LHS in combined design + uncertain 5-dimensional space. Ten runs of the method are performed to evaluate its average performance. Feasibility of the optimal solutions is validated using MPPIR method with true function evaluations. Optimal solution from nested PMA was validated using SS method, and the obtained reliability index is βSS = 2.51, which is in a good agreement with βHL = 2.50. The comparison of the method performance to the other RBDO approaches is given in Table 1. Table 1: Short column test example results. Method Nested PMA RBDO SORA Sequential EGO/EGRA21 Proposed method

Objective avg./best 216.71 216.71 217.8 (216.5) 216.74 (216.71)

Constr. viol 0.0 0.0 0.0 1.5 · 10−4

Evaluations avg. 1923 301 146.1 43.9

The proposed technique is able to find feasible optimal solution, superior to the EGO/EGRA approach, with three times less function evaluations. The validated constraint violation is less than 1.5 · 10−4 (for β values), and is comparable to the used MPPIR convergence tolerance (10−4 ). Highly non-linear limit-state function A problem, proposed originally by Lee and Jungin20 and used by Dubourg,22 tests the proposed method for the case of highly non-linear limit-state function. The problem is defined as follows: minimize (µ1 − 3.7)2 + (µ2 − 4)2 subject to: β [x2 sin(4x1 ) − 1.1x2 sin(2x2 ]) ≥ 0] ≥ 2

(11)

β [x1 + x2 − 3 ≥ 0] ≥ 2 The variable x1 and x2 are the uncertain-design variables, with their means µ1 , µ2 being design variables (µ1 ∈ [0, 3.7], µ1 ∈ [0, 4]), while having attached uncertainty. For the given µ(x1 ), µ(x2 ), x1 and x2 are 9 of 16 American Institute of Aeronautics and Astronautics

distributed normally: xi ∼ Normal(µi , σ = 0.1). Initial 11 points are generated using optimized for uniformity LHS. Ten runs of the method are performed to evaluate its average performance. The method behaviour is illustrated in Fig. 4, where the final state of original constraint and PM constraint are shown, as well as infill points, added during the optimization. Black lines represent the exact limit-state functions, the red contour - the GP surrogate for the non-linear limit-state and the green contour - the GP surrogate predicted feasible region w.r.t. specified β f index. Black points represent initial samples, green ones - points, added to the constraint surrogates only, blue points - added to both constraint and objective surrogates. Red point indicates optimal solution. Near the optimal solution the constraint is approximated very accurately, and method only adds points in the areas of possible interest.

Figure 4: Illustration of the method behaviour for test problem 2. The results summary for this test problem is given in Table 2, where βHL and βSS are the validated reliability indices using respectively FORM and SS methods for the first constraint. The second constraint is inactive for the optimal solution. In average only 37 points are evaluated in total to get the optimal solution. Table 2: Highly non-linear limit-state function test example results. Method Nested PMA RBDO SORA CBS + AMV20 Meta RBDO22 Proposed method

Obj. avg./best 1.304 1.304 1.26 1.35 1.310 (1.304)

Design point [2.82, 3.28] [2.82, 3.30] [2.82, 3.30] [2.81, 3.25] [2.82, 3.28]

βHL 2.0 2.0 1.82 >2.0 2.0

βSS 1.87 1.87 1.67 2.0 1.87

Evaluations avg. 1341 581 90 80 37

The results of the method are in perfect agreement with the nested PMA and SORA methods, as it is also based on PMA. In this sense, the method outperforms the approach of Lee and Jung, which produced infeasible solution w.r.t. both FORM and SS computed reliability indices. While the proposed method works with MPP-based reliability assessment, which provides only FORM-optimal solutions, it needs twice less true function evaluations then Meta RBDO. Application to a simplified aerodynamic optimization of the vane Aerodynamic 3D shape optimization of the LPT 2nd stage is used to test the method with the realistic simulation-based example with the possible presence of the numerical noise. Objective is to minimize entropy rise of the turbine stage, while keeping the mass flux above the given constraint. Most of the design variables are considered to be deterministic, while thickness of the leading and trailing edges are treated as uncertaindesign variables. The blade stacking line is considered to have an uncertainty in axial and circumferential 10 of 16 American Institute of Aeronautics and Astronautics

shift w.r.t. the baseline design. Normal distributions, derived for the manufacturing tolerances, are used to represent uncertainties. Mean of the objective is taken as the robust minimization goal, and reliability f index βHL ≥ 3 is required for the mass flux constraint. The summary of the problem setup, including list of design variables (in total 31), objective and constraint definitions, is given in Table 3. Table 3: Setup of the simplified aerodynamic robust shape optimization of the stage. Parameter Blade angle LE (Hub, Mid, Tip) Blade angle TE (Hub, Mid, Tip) Stagger (Hub, Mid, Tip) Wedge angle LE (Hub, Mid, Tip) Wedge angle TE (Hub, Mid, Tip) Thickness LE (Hub, Mid, Tip) Thickness TE (Hub, Mid, Tip) Stacking line shift (0,25,50,75,100%) Response Stage entropy rise Mass flux

Parameter type Deterministic design Deterministic design Deterministic design Deterministic design Deterministic design Uncertain design Uncertain design Stochastic noise Response type Objective Constraint

Definition Range Range Range Range Range Normal distribution with variable mean Normal distribution with variable mean Normal distribution Definition Minimize mean value f βHL ≥3

An initial sample of 100 points is created using optimized LHS. For the local robustness quantification, 5000 points were sampled in the uncertain variables space for each design point in order to evaluate mean value and variances of the objective function. PMA analysis is performed using the aforementioned MPPIR method, with the convergence tolerance (based on function values) equal to 1e-4. The results to the best found solution are validated with 100 true simulations. Results In total, 12 iterations were performed during the optimization, and 71 new infill points were added to the initial surrogate models. The history of the objective and constraint values, used for the surrogates fitting are shown in Fig. 5, where initial sampling and infill sampling phases can be clearly distinguished. The entropy rise mean value (objective) has been improved by 13.8% compared to the baseline design. The objective function values at the added infill points are shown in Fig. 5. The prediction accuracy is only 0.25% and is small compared to the entropy rise standard deviation.

Figure 5: Predicted objective at the infill points and validated distribution for the optimal solution. The robust constraint for the infill points is shown in Fig. 6 and is predicted to be satisfied for the optimal solution. Comparison of the predicted robust constraint value with the obtained validated value are in a good agreement, with small constraint prediction error compared to the constraint’s standard deviation.


Figure 6: Predicted robust constraint values at the infill points and validated distribution for the optimum.

Compared to the deterministic optimization results, reported in,18 where the constraint was active for the optimal solution, obtained mass flux mean value is shifted from the threshold to provide required reliability level. At the same time, this leads to approx. 0.5% higher entropy rise. With this simplified aero-optimization setup the ability of the proposed method to solve realistic FE simulation-based RBRDO problems within a limited number of simulations is shown, with only 171 runs required to find robust optimal solution in 31-dimensional combined design space.

VI.

Application to the vane cluster RBRDO

The main goal of the vane cluster design optimization is to minimize the aerodynamic losses (reduce entropy rise), while maintaining aerodynamic operating conditions and satisfying structural constraints. The same design variables, as in18 are taken. Uncertainty parameters are introduced, which account for manufacturing tolerances (uncertain-design variables), variation performance temperatures and acceleration/deceleration speed (implemented using corresponding η-curves slope manipulations). Based on available data, the correlation between TO and IDLE performance temperatures are introduced. In total, 31 design, 10 uncertain-design and 4 uncertain parameters are defined. All optimization constraints and objective are reformulated in a probabilistic way to account for the present uncertainty and listed in Table 4. Table 4: Vane cluster RBRDO setup summary. Response Mean of entropy rise Mass flux nominal value Volume (weight) Static stress Damage of each vane (1,2,3) Frequency resonance crossing

Goal 100% (objective) In range ±0.2% to baseline f Not higher than baseline with βHL ≥ 2.32 (P f ≤ 0.01) f Not higher than baseline with βHL ≥ 2.32 (P f ≤ 0.01) f Not higher than 1.0 with βHL ≥ 1.64 (P f ≤ 0.05) f Higher than 105% TO with βHL ≥ 2.32 (P f ≤ 0.01)

An initial sample of 250 points in the combined design-uncertain space is generated using optimized LHS. In total, 64 points from initial 250 resulted in simulation crashes, because of large allowed shape design parameter variations. To handle simulation failures within the surrogate-based optimization, additional GP surrogate is used for classification and prediction of failures, introduced in.18 Results The iterations were stopped manually after 11 steps, as no further objective improvement was gained over 3 iterations. In total, 285 successful infill points were added to the surrogates. At the last step, 174 points in the design space were shared among all GP models. The objective function and mass flux constraint 12 of 16 American Institute of Aeronautics and Astronautics

histories for the training points are shown in Fig. 7. The mean value of the entropy rise is reduced by 7.4%, compared to the baseline nominal design (which does not satisfy reliability requirements for the damage constraint). At the same time, the entropy reduction is approx. 3% less than for the deterministic optimal solution (results reported in18 ), which also does not satisfy damage constraint reliability. The mass flux nominal value is being well kept within the bounds.

Figure 7: Objective function and mass flux constraint values at the training points. It can be noted in Fig. 8 that value of the highest stress and resonance crossing are shifted form the deterministic constraint. Similar is valid for the maximum accumulated damage constraints, shown in Fig. 9.

Figure 8: Maximum stress and resonance crossing constraints at the training points.

Figure 9: Vane damage constraints at the training points. Now, the accuracy of the results is validated using true simulations. For the optimal design, samplingbased uncertainty analysis with 100 points is performed to evaluate the design robustness/reliability and compare with the surrogate-based predicted quantities. The predicted entropy rise mean differs from the 13 of 16 American Institute of Aeronautics and Astronautics

validated one only by 0.37%, which is less than entropy rise standard deviation (validated) equal to 0.4%. Much more challenging are the reliability constraints. For example, predicted stress constraint PM for required reliability level is safe at the level of 84% (constraint is inactive with safety margin of 16%). The f frequency crossing constraint, the PM for βHL = 2.32 is predicted to be 107.3% of TO rpm (safety margin is 2.3% only, with the constraint lower bound equal to 105% TO rpm). Corresponding CDFs obtained with sampling are shown in Fig. 10. Because only 100 points are sampled (due to high computational costs), validate CDFs should be viewed only as a rough estimation of true underlying CDFs at the tails. Additionally, the surrogate predicted PM are FORM-based and cannot be directly compared to the simulation-based probabilities, but this comparison can still provide some sort of feasibility assessment. The surrogatepredicted PM are in a good agreement with the validation results, showing that both constraints are fulfilled and being inactive (similar to the deterministic optimum).

Figure 10: Validation of the optimum w.r.t. max. stress and resonance crossing constraints’ reliability. More interesting are the damage constraints, as they are expected to be highly non-linear and active for the optimal design. The predicted damage PM for the required 95% reliability are 0.16, 0.45 and 0.11 for vanes one, two and three respectively. Sampling based CDFs for the first two vanes are shown in Fig. 11 together with the surrogate-based predicted damage levels, indicating that constraints are fulfilled.

Figure 11: Validation of the optimum w.r.t. vanes one and two damage reliability. For the 3rd , the validated reliability level is significantly lower, that the required one (87% instead of 95%), while it was predicted to be feasible. The reason for that was identified by careful examining of the actual values for the third vane damage: it is found, that two critical locations exist (trailing edge lower region and upped leading edge fillet region), where maximum damage occur, depending on the parameter values. It can be seen in Fig. 12(a), that CDF function for the vane 3 max. damage is significantly different from the two others. Also, there is no visible correlation between maximum damage values for two critical locations at the vane (see Fig. 12(b)), which might indicate different damage accumulation sources and lead to different MPFPs or MPTPs for the maximum vane damage constraint. As was mentioned earlier, in case of several MPFP, MPP-based method, and in particular employed here MPPIR method can lead to significant errors in the predicted reliability measures.


(a) Sampling-based CDFs.

(b) Damage correlation for two critical locations.

The vane cluster RBRDO problem can be reformulated, with maximum damage resultants for vanes being divided into several separate constraints, one for each of the critical locations. This most-likely will reduce the non-linearity of the limit-state functions and improve the accuracy of PMA-based results, but this is the topic of the future research. In general, the proposed method is able to solve RBRDO problem with 45 variables and 7 constraints with less than 500 true simulations.

VII.

Conclusions

The novel approach for the efficient surrogate-based reliability-based robust design optimization (RBRDO) is introduced within this research. The key idea is to combine efficient global optimization (EGO) method on the top level with the advanced performance measure analysis (PMA) for the reliability assessment. The level one Gaussian-process (GP) surrogates are constructed in the combined design-uncertain space to approximate objective and constraints. The PMA is performed for each constraint at all training design point, and obtained performance measures are used to build the level two GP surrogates, which share the points with the level one surrogate in the design space. The EGO method operates on the second level surrogates to identify promising infill points in the design space. Then, uncertain coordinates of the new infill points are found to refine the level one surrogates in the uncertain design space. New infill points are added to the training sets on the level one models and the iterations continue until convergence. The proposed method is tested using well-known analytical test problems and compared with other recent adaptive GP-based RBRDO methods. Testing results show the capability of the method to solve RBRDO problems, also in case of non-linear limit-state function with the lowest number of true design evaluations among the compared approaches. Additionally, the simplified 3D aerodynamic RBRDO problem is solved, showing the ability of the method to deal with the simulation-based black-box high-dimensional (31 variable) problems. The obtained results are compared with the deterministic optimum (reported in18 ) and validated using sampling. Finally the proposed method is applied to the vane cluster RBRDO problem. The obtained results are compared with the deterministic optimum (reported in18 ) and validated using sampling. For all except one of the constraints, good agreement with the validated reliability measures is found. For one of the vanes, the validated reliability level was lower, than the required one (87% instead of 95%). Careful analysis of the constraint indicates, that two different local damage maxima are present, driven by different problem parameters, which most likely results in complicated non-linear limit-state function with multiple MPPs. This leads to the inconsistency between PMA and sampling-based results and difference between validated and predicted reliability level. In general, the proposed method is able to solve RBRDO problem with 45 variables and 7 constraints with less than 500 true simulations. The on-going research is aimed to include more accurate PMA analysis, capable of handling multiple MPPs and more advanced GP surrogates refinement criteria for the proposed strategy is being developed in order to improve the surrogates quality over the uncertain variables space. Additionally, the vane cluster RBRDO problem will be reformulated, with maximum damage resultants for vanes being divided into several separate constraints, one for each of the critical locations, with the aim to reduce the non-linearity of the limit-state functions and improve the accuracy of PMA-based results.


Acknowledgements The authors would like to thank MTU Aero Engines and the Technische Universit¨at M¨ unchen for their support to the project. The funding of the work through the Luftfahrtforschungsprogramm 4 of the BMWi is gratefully acknowledged.

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