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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]

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]

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]

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]