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Additive manufacturing (AM) has an affinity with topology optimization to think of various designs with complex structures. Hence, this paper aims to optimize the design of a lattice-structured heat sink, which can be manufactured by AM. The design objectives are to maximize the thermal performance of convective heat transfer in natural convection simulated by computational fluid dynamics (CFD) and to minimize the material cost required for AM process at the same time. The lattice structure is represented as a node/edge system via graph theory with a moderate number of design variables. Bayesian optimization, which employs the non-dominated sorting genetic algorithm II and the Kriging surrogate model, is conducted to search for better designs with the minimum CFD cost. The present topology optimization successfully finds better lattice-structured heat sink designs than a reference fin-structured design regarding thermal performance and material cost. Also, several optimized lattice-structured designs outperform reference pin-fin-structured designs regarding thermal performance though the pin-fin structure is still advantageous for a material cost-oriented design. This paper also discusses the flow mechanism observed in the heat sink to explain how the optimized heat sink structure satisfies the competing design objectives simultaneously. [ABSTRACT FROM AUTHOR]

• Developing an efficient bi-fidelity method for uncertainty quantification of high dimensional CFD problems. • Implementing the method in uncertainty quantification of two real engineering problems. • Reducing the computational cost in both test cases in comparison to the combination of POD and full PCE methods. In the current paper, an efficient surrogate model based on combination of Proper Orthogonal Decomposition (POD) and compressed sensing is developed for affordable representation of high dimensional stochastic fields. In the developed method, instead of the full (or classical) Polynomial Chaos Expansion (PCE), the ℓ 1 -minimization approach is utilized to reduce the computational work-load of the low-fidelity calculations. To assess the model capability in the real engineering problems, two challenging high-dimensional CFD test cases namely; i) turbulent transonic flow around RAE2822 airfoil with 18 geometrical uncertainties and ii) turbulent transonic flow around NASA Rotor 37 with 3 operational and 21 geometrical uncertainties are considered. Results of Uncertainty Quantification (UQ) analysis in both test cases showed that the proposed multi-fidelity approach is able to reproduce the statistics of quantities of interest with much lower computational cost than the classical regression-based PCE method. It is shown that the combination of the POD with the compressed sensing in RAE2822 and Rotor 37 test cases gives respectively computational gains between 1.26–7.72 and 1.79–9.05 times greater than the combination of the POD with the full PCE. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the use of multiple kernel functions for assisting single-objective Kriging-based efficient global optimization (EGO). The primary objective is to improve the robustness of EGO in terms of the choice of kernel function for solving a variety of black-box optimization problems in engineering design. Specifically, three widely used kernel functions are studied, that is, Gaussian, Matérn-3/2, and Matérn-5/2 function. We investigate both model selection and ensemble techniques based on Akaike information criterion (AIC) and cross-validation error on a set of synthetic (noiseless and noisy) and non-algebraic (aerodynamic and parameter tuning) optimization problems; in addition, the use of cross-validation-based local (i.e., pointwise) ensemble is also studied. Since all the constituent surrogate models in the ensemble scheme are Kriging models, it is possible to perform EGO since the Kriging uncertainty structure is still preserved. Through analyses of empirical experiments, it is revealed that the ensemble techniques improve the robustness and performance of EGO. It is also revealed that the use of Matérn-kernels yields better results than those of the Gaussian kernel when EGO with a single kernel is considered. Furthermore, we observe that model selection methods do not yield any substantial improvement over single kernel EGO. When averaged across all types of problem (i.e., noise level, dimensionality, and synthetic/non-algebraic), the local ensemble technique achieves the best performance. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the capability of the universal Kriging (UK) model for single-objective global optimization applied within an efficient global optimization (EGO) framework. We implemented this combined UK-EGO framework and studied four variants of the UK methods, that is, a UK with a first-order polynomial, a UK with a second-order polynomial, a blind Kriging (BK) implementation from the ooDACE toolbox, and a polynomial-chaos Kriging (PCK) implementation. The UK-EGO framework with automatic trend function selection derived from the BK and PCK models works by building a UK surrogate model and then performing optimizations via expected improvement criteria on the Kriging model with the lowest leave-one-out cross-validation error. Next, we studied and compared the UK-EGO variants and standard EGO using five synthetic test functions and one aerodynamic problem. Our results show that the proper choice for the trend function through automatic feature selection can improve the optimization performance of UK-EGO relative to EGO. From our results, we found that PCK-EGO was the best variant, as it had more robust performance as compared to the rest of the UK-EGO schemes; however, total-order expansion should be used to generate the candidate trend function set for high-dimensional problems. Note that, for some test functions, the UK with predetermined polynomial trend functions performed better than that of BK and PCK, indicating that the use of automatic trend function selection does not always lead to the best quality solutions. We also found that although some variants of UK are not as globally accurate as the ordinary Kriging (OK), they can still identify better-optimized solutions due to the addition of the trend function, which helps the optimizer locate the global optimum. [ABSTRACT FROM AUTHOR]

Reducing the noise level in the acoustic cavities is the important problem when treating inflight conditions of commercial planes or boats. Shape optimization of the acoustic cavity that will take into account the geometrical and material uncertainties, arising during the manufacturing process, is presented in this paper. The noise level is controlled by minimizing the energy density in the cavity, obtained through an energy method called Simplified Energy Method. Such formulation is based on our previous published work where transformation function mapping 3D cavity surface on a 2D domain was proposed. The optimization process directly relies on this function and thus avoids remeshing of the geometry. Robust optimization is performed using the nondominated sorting genetic algorithm (NSGA-II) together with the Kriging surrogate model. Influence of geometrical and material characteristics on the optimal solution is identified. [ABSTRACT FROM AUTHOR]

The many-objective optimization performance of the Kriging-surrogate-based evolutionary algorithm (EA), which maximizes expected hypervolume improvement (EHVI) for updating the Kriging model, is investigated and compared with those using expected improvement (EI) and estimation (EST) updating criteria in this paper. Numerical experiments are conducted in 3- to 15-objective DTLZ1-7 problems. In the experiments, an exact hypervolume calculating algorithm is used for the problems with less than six objectives. On the other hand, an approximate hypervolume calculating algorithm based on Monte Carlo sampling is adopted for the problems with more objectives. The results indicate that, in the nonconstrained case, EHVI is a highly competitive updating criterion for the Kriging model and EA based many-objective optimization, especially when the test problem is complex and the number of objectives or design variables is large. [ABSTRACT FROM AUTHOR]

This paper describes the optimal flight of a discus. Two objective functions are considered. One is the flight distance, and the other is robustness. Robustness is defined as insensitivity to deviations from the local optimal release conditions. The aim of the optimization is to maximize both flight distance and robustness. There are ten control and design variables, which include variables concerned with the skill of the thrower and with the design of the equipment. It was found that there is a trade-off between flight distance and robustness. It was also found that lower drag is preferable in the first half of flight, and greater lift and drag are preferable in the latter half of flight for the world record throw of 77 m. In order to do this, the initial angle between the velocity vector and the discus planform should be negative at the launch, even if the initial lift acts vertically downward. The attitude of the discus should be inclined at launch. The discus rolls with time during the flight, so that the discus planform becomes almost in the horizontal plane in the latter half of the flight. Lower mass is good for achieving longer flight distance, but conversely, mass should be slightly greater for robustness. [ABSTRACT FROM AUTHOR]

This paper presents a comparison of the criteria for updating the Kriging surrogate models in multi-objective optimization: expected improvement (EI), expected hypervolume improvement (EHVI), estimation (EST), and those in combination (EHVI + EST). EI has been conventionally used as the criterion considering the stochastic improvement of each objective function value individually, while EHVI has recently been proposed as the criterion considering the stochastic improvement of the front of nondominated solutions in multi-objective optimization. EST is the value of each objective function estimated nonstochastically by the Kriging model without considering its uncertainties. Numerical experiments were implemented in the welded beam design problem, and empirically showed that, in an unconstrained case, EHVI maintains a balance between accuracy, spread, and uniformity in nondominated solutions for Kriging-model-based multiobjective optimization. In addition, the present experiments suggested future investigation into techniques for handling constraints with uncertainties to enhance the capability of EHVI in constrained cases. [ABSTRACT FROM AUTHOR]

Carbon nanotubes (CNTs) are novel materials with extraordinary mechanical properties. To gain insight on the design of high-mechanical-performance CNT-reinforced composites, the optimal structure of CNTs with high nominal tensile strength was determined in this study, where the nominal values correspond to the cross-sectional area of the entire specimen, including the hollow core. By using machine learning-assisted high-throughput molecular dynamics (HTMD) simulation, the relationship among the following structural parameters/properties was investigated: diameter, number of walls, chirality, and crosslink density. A database, comprising the various tensile test simulation results, was analyzed using a self-organizing map (SOM). It was observed that the influence of crosslink density on the nominal tensile strength tends to gradually decrease from the outside to the inside; generally, the crosslink density between the outermost wall and its adjacent wall is highly significant. In particular, based on our calculation conditions, five-walled, armchair-type CNTs with an outer diameter of 43.39 Å and crosslink densities (between the inner wall and outer wall) of 1.38 ± 1.16%, 1.13 ± 0.69%, 1.54 ± 0.57%, and 1.36 ± 0.35% were believed to be the optimal structure, with the nominal tensile strength and nominal Young's modulus reaching approximately 58–64 GPa and 677–698 GPa. [ABSTRACT FROM AUTHOR]

The surrogate model is an essential part of modern design optimization and exploration. In some cases, exploration of design space in multi-objective problems is important to reveal useful design insight and guidelines that will be useful for engineers. However, most surrogate models are black boxes, making interpretation difficult. This paper investigates the framework of explainable surrogate models using Shapley Additive Explanations (SHAP) to gain important design insight that helps users better understand the relationship between objective functions and design variables. We applied the explainable surrogate model framework to multi-objective design problems and performed a comparison with active subspaces and Sobol indices. Several techniques to extract design insight based on SHAP values are discussed: the averaged SHAP, the SHAP summary plot, the single- and bi-objective SHAP dependence plot, and the SHAP correlation matrix. Two aerodynamic design cases are selected to demonstrate the capability of explainable surrogate models: nine-variable inviscid and twenty-variable viscous transonic airfoil design. The findings indicate that SHAP provides more valuable insights than active subspaces and Sobol indices, particularly regarding the impact of individual design variables on the objectives. Consequently, SHAP can be employed in conjunction with active subspaces and Sobol indices to explore the input–output relationship in multi-objective design exploration comprehensively. [ABSTRACT FROM AUTHOR]

• Self-organizing map is performed on thermophysical properties of liquid substances. • Machine learning-assisted screening protocol of liquid substances is proposed. • Liquid substances are automatically categorized based on various properties. • Easy visualization technique of relative relationships among liquids is provided. In order to develop an efficient framework for global screening in the material exploration, we performed a clustering analysis of machine learning on the multi-dimensional thermophysical properties of the liquid substances. Data mining using a self-organizing map (SOM) based on the unsupervised learning was employed to project high-dimensional thermophysical data onto a low-dimensional space. Here we adopted 98 liquid substances with eight thermo-physical properties for the SOM training in order to group the liquid substances. The present SOM-clustering approach properly categorized liquid substances according to the chemical species characterized by the functional groups. [ABSTRACT FROM AUTHOR]

Multi-objective optimization in aerodynamics plays an important role in revealing trade-offs between conflicting objectives in order to discover important knowledge and insight for better future design. Of interest here is the use of Kriging surrogate models incorporated into a sequential Bayesian optimization (BO) strategy. In this paper, we studied four variants of multi-objective BO (MOBO) techniques that are based on expected improvement (EI), that is, Euclidean-based EI (EEI), expected hypervolume improvement (EHVI), ParEGO, and expected inverted penalty boundary intersection improvement (EIPBII) to understand their capabilities on handling multi-objective aerodynamic optimization problems. Numerical tests were performed on a set consisting of six generalized Schaffer problems (GSP), five low-fidelity, and one high-fidelity airfoil design problems. Results suggest that EHVI is the only method which consistently performed well on artificial and aerodynamic problems. EEI yields the worst performance and is not suitable to deal with various problem complexities. ParEGO, although it performs modestly on GSP, surprisingly works well on the low- and high-fidelity problems. On the other hand, EIPBII encounters the opposite case, where it is one of the best performer on GSP but yields modest performance on the aerodynamic problems. In light of the results, we suggest that EHVI is a highly potential MOBO method to be applied for multi-objective aerodynamic design optimization. [ABSTRACT FROM AUTHOR]

The presence of uncertainties is inevitable in engineering design and analysis, where failure in understanding their effects might lead to the structural or functional failure of the systems. The role of global sensitivity analysis in this aspect is to quantify and rank the effects of input random variables and their combinations to the variance of the random output. In problems where the use of expensive computer simulations is required, metamodels are widely used to speed up the process of global sensitivity analysis. In this paper, a multi-fidelity framework for global sensitivity analysis using polynomial chaos expansion (PCE) is presented. The goal is to accelerate the computation of Sobol sensitivity indices when the deterministic simulation is expensive and simulations with multiple levels of fidelity are available. This is especially useful in cases where a partial differential equation solver computer code is utilized to solve engineering problems. The multi-fidelity PCE is constructed by combining the low-fidelity and correction PCE. Following this step, the Sobol indices are computed using this combined PCE. The PCE coefficients for both low-fidelity and correction PCE are computed with spectral projection technique and sparse grid integration. In order to demonstrate the capability of the proposed method for sensitivity analysis, several simulations are conducted. On the aerodynamic example, the multi-fidelity approach is able to obtain an accurate value of Sobol indices with 36.66% computational cost compared to the standard single-fidelity PCE for a nearly similar accuracy. [ABSTRACT FROM AUTHOR]

Surrogate models are indispensable tools in uncertainty quantification and global sensitivity analysis. Polynomial chaos expansion (PCE) is one of the most widely used surrogate models, thanks to its faster convergence rate compared to Monte Carlo simulation. In some cases, especially for complex problems, analyzing the complexity of the random input–output relationship (e.g., nonlinearity and interactions between input variables) may reveal additional information and useful insight. To that end, this paper introduces the use of Shapley additive explanations (SHAP) to help the explanation of a PCE model. Originating from game theory and machine learning, SHAP computes the contribution of the input variables to the single prediction level. SHAP enables visual inspection of the nonlinearity and interaction between variables from a PCE model. In addition, as an alternative to Sobol indices, SHAP also quantifies the relative importance of the inputs to the output. This paper introduces a procedure to calculate SHAP values from a PCE model without explicitly building multiple PCE models. A fast and exact algorithm that enables the calculation of SHAP for high-dimensional problems is presented. The usefulness of SHAP with PCE is demonstrated on several algebraic and non-algebraic problems. • An explainable AI technique is implemented within polynomial chaos expansion (PCE). • The Shapley Additive Explanations (SHAP) enhances the explainability of PCE. • A method to analytically calculate SHAP for PCE is presented. • SHAP allows visualization of nonlinearity and interactions from the PCE model. • The method is demonstrated on three nonlinear uncertainty quantification problems. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of kernel functions on the accuracy of bi-fidelity Gaussian process regressions (GPR) for engineering applications. The potential of composite kernel learning (CKL) and model selection is also studied, aiming to ease the process of manual kernel selection. Using the autoregressive Gaussian process as the base model, this paper studies four kernel functions and their combinations: Gaussian, Matern-3/2, Matern-5/2, and Cubic. Experiments on four engineering test problems show that the best kernel is problem dependent and sometimes might be counter-intuitive, even when a large amount of low-fidelity data already aids the model. In this regard, using CKL or automatic kernel selection via cross validation and maximum likelihood can reduce the tendency to select a poor-performing kernel. In addition, the CKL technique can create a slightly more accurate model than the best-performing individual kernel. The main drawback of CKL is its significantly expensive computational cost. The results also show that, given a sufficient amount of samples, tuning the regression term is important to improve the accuracy and robustness of bi-fidelity GPR, while decreasing the importance of the proper kernel selection. [ABSTRACT FROM AUTHOR]

In this paper, to reduce the computational cost of Proper Orthogonal Decomposition (POD) method in film-cooling problems, a new method based on the combination of POD method with Sparse Polynomial Chaos Expansion (PCE) and Kriging approaches is developed. In the present method, firstly, by combining compressed sensing and POD methods, the optimal basis functions are obtained via low-fidelity calculations. These basis functions are subsequently used in the trend part of the Kriging surrogate model. Then, the trend coefficients and the stochastic Gaussian part are estimated through a limited number of high-fidelity calculations. The performance of the proposed method is investigated on two challenging test cases: 1) film-cooling jet in cross-flow and 2) leading edge film cooling of a gas turbine blade with 7 and 10 uncertain variables, respectively. For the first test case, the new method reduces the computational cost by 91 % and 53.9 % with respect to the full PCE and POD methods. For the second test case, the computational cost reductions are 81 % and 37.9 % in comparison to the full PCE and POD methods, respectively. In general, the proposed method is better in terms of accuracy in comparison to combined POD and compressed sensing, especially in the cases with the lower number of low-fidelity samples. [ABSTRACT FROM AUTHOR]

We expose an efficient strategy to deal with shape optimization of dynamical systems exhibiting flutter-type instability induced by friction, such as the considered disc-pad system. The stability of such systems can be analyzed through complex eigenvalue analysis, through which we present a squeal noise criterion to be minimized as a computationally expensive black-box function. The computational domain is discretized through isogeometric formulation for its advantages in optimization and superior approximation properties which are well studied in structural dynamics. To be computationally efficient with the expensive black-box function, we defined the optimization based on efficient global optimization scheme in the context of multi-objective optimization, with the integration of Isogeometric design-through-analysis methodology. As gradient information is hard to access for such black-box functions, in addition to the presence of constraints, we relied on meta-heuristic approach as a more generic strategy for realizing optimization of such functions in multi-objective context. As one such scheme with its own advantages was observed to provide lack of resolution to define expected improvement (EI) with a single reference value, we propose a multi-reference acquisition strategy which can be defined through a fast and efficient algorithm with fewer adaptation to the existing scheme. Results show the efficiency of this approach for our applicative example, which can be extended to other such applications as well. [ABSTRACT FROM AUTHOR]

This paper proposes a variant of gradient-enhanced surrogate model based on polynomial-chaos-kriging (PCK) to assist aerodynamic design exploration. The main aim is to improve the accuracy of kriging when gradient information is available via the adjoint method. In contrast to the standard gradient-enhanced kriging (GEK), the gradient information is used to enrich both the trend function and the stochastic part, using the polynomial-chaos expansion (PCE) as the trend function. The optimal polynomial terms are selected by scanning through various polynomial orders in which the coefficients are calculated by means of the generalized least squares or the least angle regression algorithm. The performance of the gradient-enhanced PCK (GEPCK) was compared with GEK, gradient-enhanced PCE (GEPCE), and their nongradient counterparts on a set of analytical and a suite of aerodynamic problems consisting of aerodynamic design exploration of a blended-wing-body configuration, an inviscid transonic flow over an airfoil, and a low-fidelity aerostructural wing design. The results show notable improvements obtained by GEPCK in terms of performance and robustness compared with other gradient-enhanced surrogate models. Based on these results, it can be concluded that enhancing the trend part with gradient is highly beneficial in improving the accuracy of the kriging surrogate model. [ABSTRACT FROM AUTHOR]

The stent geometrical design (e.g., inter-strut gap, length, and strut cross-section) is responsible for stent-vessel contact problems and changes in the blood flow. These changes are crucial for causing some intravascular abnormalities such as vessel wall injury and restenosis. Therefore, structural optimization of stent design is necessary to find the optimal stent geometry design. In this study, we performed a multiobjective stent optimization for minimization of average stress and low wall shear stress ratio while considering the wall deformation in 3D flow simulations of triangular and rectangular struts. Surrogate-based optimization with Kriging method and expected hypervolume improvement (EHVI) are performed to construct the surrogate model map and find the best configuration of inter-strut gap (G) and side length (SL). In light of the results, G-SL configurations of 2.81-0.39 and 3.00-0.43 mm are suggested as the best configuration for rectangular and triangular struts, respectively. Moreover, considering the surrogate model and flow pattern conditions, we concluded that triangular struts work better to improve the intravascular hemodynamics. ᅟ Graphical abstract. [ABSTRACT FROM AUTHOR]

Optimal non-planar wing planform geometries for a Mach 2.2 45-seat supersonic transport are generated using Kriging-based Bayesian optimisation techniques on a 22-variable, 3-objective problem for two wing-loading conditions. Sonic boom loudness, drag, and root bending moment are simultaneously minimised at a fixed lift. A Python-based framework is developed for rapid evaluation of designs in a parallel computation environment, automating conceptual geometry generation, optimised mesh griding, Euler CFD evaluation, and thermoviscous atmospheric boom propagation. An empirical turbulent skin friction coefficient is used to correct the Euler drag and an estimate of wave drag is performed. Self-organising maps and polynomial chaos expansion Shapley effects are used to explore the design variables and their associated sensitivities with respect to the design objectives. A geometric centroid-based proxy for trim is used to filter the results; feasible extreme designs and a compromise design are compared and discussed. The results indicate a trade-off between each of the three objectives due to competing design requirements, however an improvement in two of the objective functions is possible by sacrificing the remaining objective. One possible compromise solution reduces the root bending moment coefficient by 25% and the sonic boom loudness by 24% at the cost of a 5% increase in drag. [ABSTRACT FROM AUTHOR]