This paper presents a methodology for constrained efficient global optimization (EGO) using support vector machines (SVMs). While the objective function is approximated using Kriging, as in the original EGO formulation, the boundary of the feasible domain is approximated explicitly as a function of the design variables using an SVM. Because SVM is a classification approach and does not involve response approximations, this approach alleviates issues due to discontinuous or binary responses. More importantly, several constraints, even correlated, can be represented using one unique SVM, thus considerably simplifying constrained problems. In order to account for constraints, this paper introduces an SVM-based "probability of feasibility" using a new Probabilistic SVM model. The proposed optimization scheme is constituted of two levels. In a first stage, a global search for the optimal solution is performed based on the "expected improvement" of the objective function and the probability of feasibility. In a second stage, the SVM boundary is locally refined using an adaptive sampling scheme. An unconstrained and a constrained formulation of the optimization problem are presented and compared. Several analytical examples are used to test the formulations. In particular, a problem with 99 constraints and an aeroelasticity problem with binary A. Basudhar · C. Dribusch · S. Lacaze · S. Missoum (B) Aerospace and Mechanical Engineering Department, output are presented. Overall, the results indicate that the constrained formulation is more robust and efficient.
This paper introduces a methodology for the reliability-based design optimization of systems with nonlinear aeroelastic constraints. The approach is based on the construction of explicit flutter and subcritical limit cycle oscillation boundaries in terms of deterministic and random design variables. The boundaries are constructed using a support vector machine that provides a way to efficiently evaluate probabilities of failure and solve the reliabilitybased design optimization problem. Another major advantage of the approach is that it efficiently manages the discontinuities that might appear during subcritical limit cycle oscillations. The proposed approach is applied to the construction of flutter and subcritical limit cycle oscillation boundaries for a two-degree-of-freedom airfoil with nonlinear stiffnesses. The solution of a reliability-based design optimization problem with a constraint on the probability of subcritical limit cycle oscillation is also provided.
The Giant Magellan Telescope (GMT) is one of Extremely large telescopes, which is 25m in diameter featured with two Gregorian secondary mirrors, an adaptive secondary mirror (ASM) and a fast-steering secondary mirror (FSM). The FSM is 3.2 m in diameter and built as seven 1.1 m diameter circular segments conjugated 1:1 to the seven 8.4 m segments of the primary. The guiding philosophy in the design of the FSM segment mirror is to minimize development and fabrication risks ensuring a set of secondary mirrors are available on schedule for telescope commissioning and early operations in a seeing limited mode. Each FSM segment contains a tip-tilt capability for fine co-alignment of the telescope sub-apertures and fast guiding to attenuate telescope wind shake and mount control jitter, thus optimizing the seeing limited performance of the telescope. The final design of the FSM mirror and support system configuration was optimized using finite element analyses and optical performance analyses. The optical surface deformations, image qualities, and structure functions for the gravity print-through cases, thermal gradient effects, and dynamic performances were evaluated. The results indicated that the GMT FSM mirror and its support system will favorably meet the optical performance goals for residual surface error and the FSM surface figure accuracy requirement defined by encircled energy (EE80) in the focal plane. The mirror cell assembly analysis indicated an excellent dynamic stiffness which will support the goal of tip-tilt operation.
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