Surrogate-based Robust Airfoil Optimization under Aleatory Flight Condition and Geometric Uncertainties

A robust topology optimization algorithm is proposed for frame structures in the presence of geometric or material properties uncertainties. While geometric uncertainties were modeled with uncorrelated random variables expressing the node locations of the structure, material properties uncertainties were modeled with a correlated random field of the material Young’s modulus with an exponentially decaying correlation structure throughout the domain. The proposed algorithm uses stochastic perturbation method for propagating these uncertainties to the structural response level, measured in terms of compliance, and optimizes the expected value plus multiple factors of the standard deviation of the response. A comparison between the resulting robust designs and deterministic designs is made, and changes to the final topologies are discussed. Moreover, using Monte Carlo simulation, it was shown that the robust designs outperform the deterministic designs under real-world situations that are accompanied with uncertainties.

The importance of designing airfoils to be robust with respect to uncertainties in operating conditions is well recognized. However, often, the probability distribution of such uncertainties does not exist or is unknown, and a designer looking to perform a robust optimization is tasked with deciding how to represent these uncertainties within the optimization framework. This paper asks the following: “How important is the choice of how to represent input uncertainties mathematically in robust airfoil optimization?” It specifically compares probabilistically based aleatory uncertainties and interval-based epistemic uncertainties. This is first investigated by considering optimizations on several algebraic test problems, which illustrate the mechanisms by which the representation of uncertainty becomes significant in a robust optimization. This insight is then used to predict and subsequently demonstrate that, for two airfoil design problems, the advantage of doing a robust optimization over a deterministic optimization is similar, regardless of how the input uncertainties are represented mathematically. The benefit of this is potentially eliminating the time required to establish an accurate representation of the uncertainties from the preliminary stage of design, where time is a valuable resource.

Purpose:Respiratory motion and patient setup error both contribute to the dosimetric uncertainty in radiotherapy of lung tumors. Managing these uncertainties for free-breathing treatments is usually done by margin-based approaches or robust optimization. However, breathing motion can be irregular and concerns have been raised for the robustness of the treatment plans. We have previously reported the dosimetric effects of the respiratory motion, without setup uncertainties, in lung tumor photon radiotherapy using free-breathing images. In this study, we include setup uncertainty. Methods:Tumor positions from cine-CT images acquired in free-breathing were combined with per-fraction patient shifts to simulate treatment scenarios. A total of 14 patients with 300 tumor positions were used to evaluate treatment plans based on 4DCT. Four planning methods aiming at delivering 54 Gy as median tumor dose in three fractions were compared. The planning methods were denoted robust 4D (RB4), isodose to the PTV with a central higher dose (ISD), the ISD method normalized to the intended median tumor dose (IRN) and homogeneous fluence to the PTV (FLU). Results:For all planning methods 95% of the intended dose was achieved with at least 90% probability with RB4 and FLU having equal CTV D50% values at this probability. FLU gave the most consistent results in terms of CTV D50% spread and dose homogeneity. Conclusions:Despite the simulated patient shifts and tumor motions being larger than observed in the 4DCTs the dosimetric impact was suggested to be small. RB4 or FLU are recommended for the planning of free-breathing treatments.

In order to improve the aerodynamic stability of a two-dimensional airfoil with deformable trailing edge during the change of external conditions,a robust optimization method considering the uncertainty was proposed. Based on class-shape function transformation( CST) method,a parametric model was established to represent the geometry of a two-dimensional airfoil with a deformable trailing edge. The differences between robust optimization method and deterministic optimization method were discussed. Taking into account the uncertain airfoil geometry and inlet Mach number,the robust optimization method was applied to maximize the mean value of the lift-to-drag ratio and minimize its standard deviation. The actuation power requirement of the optimized airfoil with a deformable trailing edge was calculated. The results show that the robust optimization method can help to improve the aerodynamic performance of morphing airfoil and reduce the sensitivity of this performance to inlet Mach number simultaneously and to reduce the actuation power requirement of the robust optimal airfoil.

:A robust multi-objective wing design optimization procedure using CFD is presented in this work. Instead of directly applying CFD to wing design optimization, a Kriging aerodynamic model that approximates CFD result is proposed to save computational cost. By introducing 6σ robust approach as a sub-objective function, multi-objective wing design problems can be solved with aerodynamic robustness. Non-Uniform Rational B-Spline curves (NURBS) are introduced to depict the wing geometry in design optimization process. The most important parameters of the wing geometry can then be screened by design of experiment and treated as design variables due to the local variation property of NURBS. Under this formulation, a multi-objective genetic algorithm based on non-dominated sorting is employed to handle multiple flight conditions in wing design. The optimum wing geometry is selected according to trade-off among design objects on the non-dominated front. To validate the proposed approach, two robust design optimization cases are studied for ONERA-M6-Wing. It turns out that the drag coefficient is insensitive to Mach number between Ma0.8–Ma0.9 after robust optimization. The result also verifies the effectiveness of our method in boosting performance and robustness of multiple flight conditions: the lift curve slope (Ma0.3) of optimum wing and the 6σ objective function of transonic drag (Ma0.8–Ma0.9) increase by 11.9% and 25.4%, respectively. Observations in the optimization cases are concluded as follows: 1) For wing aerodynamic design problems, the multi-objective robust optimization can both improve the performance at different flight conditions and provide robustness. 2) The Kriging aerodynamic model derived from CFD result can satisfy the precision requirement of wing design. 3) Even though the optimization result is subject to the weights of the 6σ function in sub-object, the influence is not comparable to the trade-off among design objects. 4) The optimum solution obtained by proposed approach is superior to gradient based optimization method, while the computational cost is acceptable.

Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization

Robust optimization generates scenario‐based plans by a minimax optimization method to find optimal scenario for the trade‐off between target coverage robustness and organ‐at‐risk (OAR) sparing. In this study, 20 lung cancer patients with tumors located at various anatomical regions within the lungs were selected and robust optimization photon treatment plans including intensity modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT) plans were generated. The plan robustness was analyzed using perturbed doses with setup error boundary of ±3 mm in anterior/posterior (AP), ±3 mm in left/right (LR), and ±5 mm in inferior/superior (IS) directions from isocenter. Perturbed doses for D99, D98, and D95 were computed from six shifted isocenter plans to evaluate plan robustness. Dosimetric study was performed to compare the internal target volume‐based robust optimization plans (ITV‐IMRT and ITV‐VMAT) and conventional PTV margin‐based plans (PTV‐IMRT and PTV‐VMAT). The dosimetric comparison parameters were: ITV target mean dose (Dmean), R95(D95/Dprescription), Paddick's conformity index (CI), homogeneity index (HI), monitor unit (MU), and OAR doses including lung (Dmean, V20 Gy and V15 Gy), chest wall, heart, esophagus, and maximum cord doses. A comparison of optimization results showed the robust optimization plan had better ITV dose coverage, better CI, worse HI, and lower OAR doses than conventional PTV margin‐based plans. Plan robustness evaluation showed that the perturbed doses of D99, D98, and D95 were all satisfied at least 99% of the ITV to received 95% of prescription doses. It was also observed that PTV margin‐based plans had higher MU than robust optimization plans. The results also showed robust optimization can generate plans that offer increased OAR sparing, especially for normal lungs and OARs near or abutting the target. Weak correlation was found between normal lung dose and target size, and no other correlation was observed in this study.

The importance of designing airfoils to be robust with respect to uncertainties in operating conditions is well recognized. However, often the probability distribution of such uncertainties does not exist or is unknown, and a designer looking to perform a robust optimization is tasked with deciding how to represent these uncertainties within the optimization framework. This paper asks ``how important is the choice of how to represent input uncertainties mathematically in robust airfoil optimization?", specifically comparing probabilistically based aleatory uncertainties and interval based epistemic uncertainties. This is first investigated by considering optimizations on several algebraic test problems, which illustrate the mechanisms by which the representation of uncertainty becomes significant in a robust optimization. This insight is then used to predict and subsequently demonstrate that for two airfoil design problems the advantage of doing a robust optimization over a deterministic optimization is similar regardless of how the input uncertainties are represented mathematically. The benefit of this is potentially eliminating the time required to establish an accurate representation of the uncertainties from the preliminary stage of design, where time is a valuable resource.

The classic approach in turbomachinery design optimization considers only nominal geometries while manufacturing tolerances are evaluated in post-processing. Without any knowledge about such deviations, the optimizer chooses the solution corresponding to the highest attainable performance. However, such a shape may require tight and expensive tolerances to maintain the expected performance in large-scale populations or even necessitate tolerances not realizable with the available manufacturing process. Therefore, the entire optimization activity would need to be re-run to look for a more robust solution. In contrast, Uncertainty Quantification (UQ) methods take into account the tolerances applicable to every design parameter and propagate those uncertainties to the output. When applied in combination with an optimizer, performance data along with information related to the robustness and reliability of every geometry become available, thus allowing to find optimal solutions that are less sensitive to the design parameters deviations. In this work, we present the development of an efficient evaluation framework through the application of the Probabilistic Collocation method, a sampling-based UQ technique generating high-fidelity statistical data of the uncertainties impact on the output solution, coupled with the Smolyak algorithm to reduce the sampling size. A metamodel-assisted approach steers the optimization using surrogate modeling to rapidly approximate the statistical data using the Monte-Carlo method. The UQ model is invoked only when the optimizer approaches convergence in order to confirm the robustness of the identified optimum or to lead the optimization to a more robust region of the design space. The methodology is applied to the robust optimization of a radial turbine rotor with 32 design variables, nine of which were affected by uncertainties. The proposed approach provides a more robust and reliable design against geometrical uncertainties, in addition to the sensitivity analysis, with an estimated overhead below 30% of the cost of a traditional design optimization.

Robust Airfoil Optimization Using a Conditional Expected Maximum Value Approach

Deterministic engineering design often leads to unexpected or physically unrealizable results. This is due to the fact that deterministic design is not able to capture the effects of even slight natural fluctuations of parameters. Deterministic transonic shape optimization is no exception: deterministic designs can result in dramatically inferior performance when the actual operating conditions are different from the design conditions used during a deterministic optimization procedure. The goal of this paper is to overcome the off-design performance degradation of deterministic transonic shape optimization by using two different optimization approaches to produce robust designs. Two criteria, the well-known maximum/minimum expected value criterion (MEV) and the alternative expected maximum/minimum value criterion (EMV), are studied and applied to improve an initial RAE 2822 design. It turns out that EMV is much easier to implement than MEV, given a deterministic optimization code, and may provide a promising method for optimizing design shapes under uncertainty.

Adaptive Strategy for External Beam Radiation Therapy in Prostate Cancer: Management of the Geometrical Uncertainties With Robust Optimization

Purpose:To provide a method for quantifying centre‐ and tumour site‐specific population range uncertainty due to inter‐fraction motion for use within robust proton planning and plan analysis. PTV‐based and probabilistic/robust plans depend upon accurate knowledge of geometric and range uncertainties. If the uncertainty model is inaccurate, both types of plan, PTV or probabilistic, could produce under‐dosing of the target and/or overdosing of OAR.Methods:Daily mega‐voltage volumetric imaging data from previously treated radiotherapy patients has been used to investigate inter‐fraction changes to water equivalent path‐length (WEPL). Daily image‐guidance scans were carried out for each patient and corrected for changes in CTV position (using rigid transformations). An effective depth algorithm was used to determine residual range changes, after corrections were applied, by comparing WEPL within the CTV at each fraction for several beam angles. This method has been demonstrated for nine prostate and seven head and neck.Results:Inter‐fraction range changes for the head and neck patient sample were Σ=3.39mm, σ=4.72mm and overall mean =−1.82mm; for prostate Σ=5.64mm, σ=5.91mm and overall mean =0.98 mm. The choice of beam angle for head and neck did not affect the inter‐fraction range error significantly; however this was not the same for prostate. Greater range changes were seen using a lateral beam compared to an anterior beam for prostate due to relative motion of the prostate and femoral heads.Conclusion:A method has been developed to quantify population range changes due to inter‐fraction motion. The results show a non‐zero mean for these samples, as well as significant Σ and σ, highlighting the importance of quantifying uncertainties and use of image guidance. Such information can be used in robust optimisation algorithms and for analysing treatment plan robustness. It may also help in establishing beam start conditions at planning and for establishing adaptive planning protocols.

Robust optimal operation of continuous catalytic reforming process under feedstock uncertainty

This chapter describes the use of a multiobjective memetic algorithm in aerodynamic shape optimization applications using computational fluid dynamics(CFD). A systematic overviewon multiobjective optimization is given. It covers parameterized scalarization, population-based Pareto front forming using evolutionary algorithms, formulation of Pareto front improvement using local search methods, and various possible heuristics that can be applied in formulating efficient memetic algorithms. A constrained local search, combined with a non-dominated sorting genetic algorithm with niching in both objective and design space is proposed in the paper. Test function results are given and finally the method is used to solve the robust airfoil design problem using a multi-point formulation.