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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]

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]

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]

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]