“…It is done by solving a likelihood optimisation problem. For more details about the theoretical background, the readers are invited to refer to Sacks et al (1989), Paulson and Ragkousis (2015), Jones et al (1998), Kleijnen (2017) and Cheng et al (2020). In the present study, the Kriging meta-models are build based on the ordinary Kriging using pyKriging which is an open source Python Kriging toolkit (Paulson and Ragkousis 2015).…”
Section: Kriging: Some Fundamentalsmentioning
confidence: 99%
“…The number 100 has been chosen based on a convergence study, not shown here for the sake of consistency. This sample set includes 27 points located on the boundary of the three-dimensional design space and 77 points generated using a Latin hypercube sampling through the pyKriging package (Paulson and Ragkousis 2015).…”
Section: Construction and Validation Of The Kriging Meta-modelsmentioning
confidence: 99%
“…A surrogate model, also called meta-model, has the following basic idea: from a small number of evaluations of an expensive model, a mathematical function which gives the same output for the same inputs is constructed (Cheng et al 2020). Different methods allow to do so as Kriging (Sacks et al 1989;Paulson and Ragkousis 2015), polynomial chaos (Wiener 1938;Xiu and Karniadakis 2002;Totis and Sortino 2020), support vector regression (Smola and Schölkopf 2004), etc. Using surrogate models has proven to be efficient in different cases for the design of friction dampers (Yuan et al 2020;Denimal et al 2021a).…”
Integrally bladed disks (blisk) have been widely used in the turbo-machinery industry due to its high aerodynamic performance and structural efficiency. A friction ring damper (FRD) is usually integrated in the system to improve its low damping. However, the design of the geometry of this FRD become complex and computationally expensive due to the strong nonlinearities from friction interfaces. In this work, we propose an efficient modelling strategy based on advanced nonlinear modal analysis and Kriging surrogate models to design and optimize the geometry of a 3D FRD attached to a high fidelity full-scale blisk. The 3D ring damper is parametrised with a few key geometrical parameters. The impact of each geometric parameter and their sensitivities to nonlinear dynamic response can be efficiently assessed using Kriging meta-modelling based on a few damped nonlinear normal modes. Results demonstrate that the damping performances of ring dampers can be substantially optimized through the proposed modelling strategy whilst key insights for the design of the rings are given. It is also demonstrated that the distribution of the contact normal load on the contact interfaces has a strong influence on the damping performances and can be effectively tuned via the upper surface geometry of the ring dampers.
“…It is done by solving a likelihood optimisation problem. For more details about the theoretical background, the readers are invited to refer to Sacks et al (1989), Paulson and Ragkousis (2015), Jones et al (1998), Kleijnen (2017) and Cheng et al (2020). In the present study, the Kriging meta-models are build based on the ordinary Kriging using pyKriging which is an open source Python Kriging toolkit (Paulson and Ragkousis 2015).…”
Section: Kriging: Some Fundamentalsmentioning
confidence: 99%
“…The number 100 has been chosen based on a convergence study, not shown here for the sake of consistency. This sample set includes 27 points located on the boundary of the three-dimensional design space and 77 points generated using a Latin hypercube sampling through the pyKriging package (Paulson and Ragkousis 2015).…”
Section: Construction and Validation Of The Kriging Meta-modelsmentioning
confidence: 99%
“…A surrogate model, also called meta-model, has the following basic idea: from a small number of evaluations of an expensive model, a mathematical function which gives the same output for the same inputs is constructed (Cheng et al 2020). Different methods allow to do so as Kriging (Sacks et al 1989;Paulson and Ragkousis 2015), polynomial chaos (Wiener 1938;Xiu and Karniadakis 2002;Totis and Sortino 2020), support vector regression (Smola and Schölkopf 2004), etc. Using surrogate models has proven to be efficient in different cases for the design of friction dampers (Yuan et al 2020;Denimal et al 2021a).…”
Integrally bladed disks (blisk) have been widely used in the turbo-machinery industry due to its high aerodynamic performance and structural efficiency. A friction ring damper (FRD) is usually integrated in the system to improve its low damping. However, the design of the geometry of this FRD become complex and computationally expensive due to the strong nonlinearities from friction interfaces. In this work, we propose an efficient modelling strategy based on advanced nonlinear modal analysis and Kriging surrogate models to design and optimize the geometry of a 3D FRD attached to a high fidelity full-scale blisk. The 3D ring damper is parametrised with a few key geometrical parameters. The impact of each geometric parameter and their sensitivities to nonlinear dynamic response can be efficiently assessed using Kriging meta-modelling based on a few damped nonlinear normal modes. Results demonstrate that the damping performances of ring dampers can be substantially optimized through the proposed modelling strategy whilst key insights for the design of the rings are given. It is also demonstrated that the distribution of the contact normal load on the contact interfaces has a strong influence on the damping performances and can be effectively tuned via the upper surface geometry of the ring dampers.
“…The Kriging model is realized via Python package pyKriging. 61 Figure 9.Eigenvector situation for different sampling seed numbers where the horizontal axis refers to total pressure ratio and the vertical axis efficiency.…”
Compressor blade design influences aero-engine performance mainly through its total pressure ratio and efficiency. As a volume-based geometric parameterization, Free-form Deformation (FFD) brings three-dimensionality that is essential to blade design. However, the manipulation of control points with respect to simple numeric parametric perturbation renders the use of design space low-efficient. Therefore, an improved FFD with ‘wiser’ control point lay-out is expected to identify those more important design variables. This paper proposes novel design variables that are assigned to grouping of control points’ displacement in FFD lay-out. In short, the approach is realized as: (i) establish a library of sufficient blade shape samples; (ii) filter the database with geometric constraints; (iii) extract dominant modes via (POD) Proper Orthogonal Decomposition; (iv) construct new design variables and apply them in optimization. With geometric constraint filtering, problem-oriented information is injected. With POD, the dominance of new selected geometric parameters in problem description is assured. Perfunctory details of displacement data of each control point in the lay-out can be replaced by grouped data as new design variable candidates. As a proof-of-concept study of the new approach, compressor blade Rotor 37 is selected to be the good platform of testing the feasibility of POD-aided FFD as a global and flexible yet economic geometric parameterization. Result demonstrates the feasibility of proposed POD-aided FFD approach that helps conduct an optimization involving displacement of 6 × 4 × 3 control points with as few as five new design variables, while still being capable of bringing optimization effect in three test cases.
“…scikit-learn provides a python-based, machine-learning-oriented implementation of Gaussian processes for regression and classification (Pedregosa et al 2011). Alternatively, PyKriging (Paulson & Ragkousis 2015) offers a Kriging toolbox in python that offers basic functionality with focus on user-friendliness. Gpy (GPy 2012) offers a Gaussian process framework with focus on regression and classification problems.…”
We introduce the Gaussian process (GP) modelling module developed within the UQLab software framework. The novel design of the GP-module aims at providing seamless integration of GP modelling into any uncertainty quantification workflow, as well as a standalone surrogate modelling tool. We first briefly present the key mathematical tools at the basis of GP modelling (a.k.a. Kriging), as well as the associated theoretical and computational framework. We then provide an extensive overview of the available features of the software and demonstrate its flexibility and user-friendliness. Finally, we showcase the usage and the performance of the software on several applications borrowed from different fields of engineering. These include a basic surrogate of a well-known analytical benchmark function, a hierarchical Kriging example applied to wind turbine aero-servo-elastic simulations and a more complex geotechnical example that requires a non-stationary, user-defined correlation function. The GP-module, like the rest of the scientific code that is shipped with UQLab, is open source (BSD license).
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