Christian Gogu

Making the Most Out of Surrogate Models: Tricks of the Trade

Design analysis and optimization based on high-fidelity computer experiments is commonly expensive. Surrogate modeling is often the tool of choice for reducing the computational burden. However, even after years of intensive research, surrogate modeling still involves a struggle to achieve maximum accuracy within limited resources. This work summarizes advanced and yet simple statistical tools that help. We focus on four techniques with increasing popularity in the design automation community: (i) screening and variable reduction in both the input and the output spaces, (ii) simultaneous use of multiple surrogates, (iii) sequential sampling and optimization, and (iv) conservative estimators.

(Student Paper) Analysis and Design of Corrugated-Core Sandwich Panels for Thermal Protection Systems of Space Vehicles

A preliminary design process of an integral thermal protection system (ITPS) has been presented. Unlike the conventional TPS, the ITPS has both thermal protection as well as load bearing capabilities. The objective of this research work is to establish procedures and identify issues in the design of an ITPS. Corrugated-core sandwich construction has been chosen as a candidate structure for this design problem. An optimization problem was formulated as part of the design process with mass per unit area of the ITPS as the objective function and different functions of the ITPS as constraints. The optimization problem was solved by developing response surface approximations to represent the constraints. Response surface approximations were obtained from finite element (FE) analyses, which include transient heat transfer analyses and buckling analyses. A Matlab code (ITPS Optimizer) has been developed for generating the response surfaces, which has the capability to carry out hundreds of FE analyses, automatically, in conjunction with ABAQUS. Accurate response surface approximations could be obtained for the peak temperatures of the ITPS structure. It was found that response surface approximations for the smallest buckling eigen value of the whole structure were inaccurate. Therefore, the buckling modes were separated and similar buckling modes were grouped together. One response surface approximation was obtained for the smallest buckling eigen value of each group. The preliminary design process for the ITPS generates a design with areal density of approximately 10 lb/ft. Even though the ITPS has much higher load bearing capabilities, it is still on the heavier side when compared to conventional TPS (typical weight 2 lb/ft). New design changes have been proposed as part of the future work to make the ITPS lighter than the current design.

Comparison of Materials for an Integrated Thermal Protection System for Spacecraft Reentry

An integrated thermal protection system for spacecraft reentry based on a corrugated core sandwich panel concept fulfilling both thermal and structural functions is optimized for minimal mass. We seek the optimal dimensions and the best materials, but directly optimizing both continuous geometric parameters and discrete material choices is difficult. Accordingly the optimization problem is solved in two steps. In the first step, good candidatematerialsareselectedbasedmainlyontheirthermalperformance, obtainedfromasplineinterpolation of the maximum bottom face sheet temperature. Mild simplifying assumptions allowed a reduction of the number of variables in the interpolation to two nondimensional variables. In combination with a material database, this procedure allowed a graphical comparison and selection of candidate materials. In the second step, the geometry of the integrated thermal protection system panel is optimized for different combinations of the materials identified in step one. The optimization considers both thermal and structural constraints. The lightest panel employs aluminosilicate/Nextel 720 composites for the top face sheet and web corrugation and beryllium for the bottom face sheet. For the same thermal reentry environment, this design was found to be only about 40% heavier than a reference conventional thermal protection system that does not provide any structural load carrying capabilities.

Dimensionality Reduction Approach for Response Surface Approximations: Application to Thermal Design

of the maximum bottom face temperature is needed. The finite element model used to evaluate the maximum temperature depended on 15 parameters of interest for the design. A small number of assumptions simplified the thermal equations, allowing easy nondimensionalization, which together with a global sensitivity analysis showed that the maximum temperature mainly depends on only two nondimensional parameters. These were selected to be the variables of the response surface approximation for maximum temperature, which was constructed using simulations from the original nonsimplified finite element model. The major error in the two-dimensional response surface approximation was found to be due to the fact that the two nondimensional variables account for only part (albeit the major part) of the dependence on the original 15 variables. This error was checked and reasonable agreement was found. The two-dimensional nature of the response surface approximations allowed graphical representation, which we used for material selection from among hundreds of possible materials for the design optimization of an integrated thermal protection system panel.

Aeroassisted Orbital Transfer Trajectory Optimization Considering Thermal Protection System Mass

Aeroassisted orbital transfer is recognized as a potential technology to enhance operational responsiveness in space with significant fuel savings. To endure aerodynamic heating resulting from the flight through the atmosphere, however, considerable thermal protection is required, thereby potentially decreasing the mass savings due to lower fuel consumption. In this paper, the relationship between achievable fuel savings and thermal protection system size is investigated by coupling trajectory optimization and thermal protection system design through the single parameter of maximum heating rate constraint. The optimal solution that minimizes the total mass of fuel and the thermal protection system is then determined. A trajectory optimization procedure and a thermal protection system mass estimation model are then applied to the transfer of a vehicle between two low-Earth orbits with a specified inclination change. All trajectory parameters, including deorbit, boost, and recircularization impulses, are optimized and the thermal protection system is sized with ablative and reusable materials. It is found that the minimum overall mass (fuel and thermal protection system) is achieved when no heating rate constraint is imposed, which is also the scenario that minimizes the fuel consumption alone.

Introduction to the Bayesian Approach Applied to Elastic Constants Identification

The basic formulation of the least-squares method, based on the L2 norm of the residuals, is still widely used today for identifying elastic constants of aerospace materials from experimental data. While this method often works well, methods that can benefit from statistical information, such as the Bayesian method, may sometimes be more accurate. We seek situations with significant difference between the material properties identified by the two methods. For a simple three-bar truss example we illustrate three situations in which the Bayesian approach systematically leads to more accurate results: different sensitivity of the measured response to the parameters to be identified, different uncertainty in the measurements, and correlation among response components. When all three effects add up, the Bayesian approach can be much more accurate. Furthermore, the Bayesian approach has the additional advantage of providing the uncertainty in the identified parameters.We also compare the two methods for a more realistic problem of identification of elastic constants from natural frequencies of a composite plate.

Optimization Based Algorithms for Uncertainty Propagation Through Functions With Multidimensional Output Within Evidence Theory

Evidence theory is one of the approaches designed specifically for dealing with epistemic uncertainty. This type of uncertainty modeling is often useful at preliminary design stages where the uncertainty related to lack of knowledge is the highest. While multiple approaches for propagating epistemic uncertainty through one-dimensional functions have been proposed, propagation through functions having a multidimensional output that need to be considered at once received less attention. Such propagation is particularly important when the multiple function outputs are not independent, which frequently occurs in real world problems. The present paper proposes an approach for calculating belief and plausibility measures by uncertainty propagation through functions with multidimensional, nonindependent output by formulating the problem as one-dimensional optimization problems in spite of the multidimensionality of the output. A general formulation is first presented followed by two special cases where the multidimensional function is convex and where it is linear over each focal element. An analytical example first illustrates the importance of considering all the function outputs at once when these are not independent. Then, an application example to preliminary design of a propeller aircraft then illustrates the proposed algorithm for a convex function. An approximate solution found to be almost identical to the exact solution is also obtained for this problem by linearizing the previous convex function over each focal element.

Multi-Fidelity Design of an Integrated Thermal Protection System for Spacecraft Reentry

For space vehicles traveling at hypersonic speeds through a planetary atmosphere, the aerodynamic forces may result in high aerodynamic heating. A Thermal Protection System (TPS) protects the vehicle structure from damage due to this heating. TPS occupies a large acreage on vehicle exteriors and accounts for a significant part of the launch weight. One potential way of saving weight is to have a load-bearing TPS that performs some structural functions. One such concept called the Integrated Thermal Protection System (ITPS) is a corrugated-core sandwich structure. A previous study on this corrugated core sandwich panel developed an optimization procedure for finding the optimal geometry leading to low mass design. The procedure used finite element analyses to construct response surface approximations (RSA) of the critical constraints, including maximum bottom facesheet temperature, maximum deflection, maximum stresses and buckling loads. The RSAs obtained are of high fidelity, however they require too much computational time. In parallel, an analytical model of the ITPS was developed based on homogenization of the panel for calculating displacements and stresses. This model, though relatively inexpensive, is not accurate enough for design optimization. In the present paper we combine the two models into a multi-fidelity model, which can be used for the design optimization of the ITPS panel. We fit the low fidelity analytical model with a high quality surrogate, which is then corrected by the use of a small number of high fidelity finite element analyses. Fitting the difference or the ratio between the high fidelity analyses and the low fidelity surrogate with a response surface approximation allows constructing a so-called correction response surface. The approach is known as multi-fidelity or variable-complexity modeling, and usually allows using significantly fewer high fidelity analyses for given accuracy.

Bayesian Identification of Elastic Constants in Multi-Directional Laminate from Moiré Interferometry Displacement Fields

The ply elastic constants needed for classical lamination theory analysis of multi-directional laminates may differ from those obtained from unidirectional laminates because of three dimensional effects. In addition, the unidirectional laminates may not be available for testing. In such cases, full-field displacement measurements offer the potential of identifying several material properties simultaneously. For that, it is desirable to create complex displacement fields that are strongly influenced by all the elastic constants. In this work, we explore the potential of using a laminated plate with an open-hole under traction loading to achieve that and identify all four ply elastic constants (E1, E2, ν12, G12) at once. However, the accuracy of the identified properties may not be as good as properties measured from individual tests due to the complexity of the experiment, the relative insensitivity of the measured quantities to some of the properties and the various possible sources of uncertainty. It is thus important to quantify the uncertainty (or confidence) with which these properties are identified. Here, Bayesian identification is used for this purpose, because it can readily model all the uncertainties in the analysis and measurements, and because it provides the full coupled probability distribution of the identified material properties. In addition, it offers the potential to combine properties identified based on substantially different experiments. The full-field measurement is obtained by moiré interferometry. For computational efficiency the Bayesian approach was applied to a proper orthogonal decomposition (POD) of the displacement fields. The analysis showed that the four orthotropic elastic constants are determined with quite different confidence levels as well as with significant correlation. Comparison with manufacturing specifications showed substantial difference in one constant, and this conclusion agreed with earlier measurement of that constant by a traditional four-point bending test. It is possible that the POD approach did not take full advantage of the copious data provided by the full field measurements, and for that reason that data is provided for others to use (as on line material attached to the article).

Skipping unnecessary structural airframe maintenance using an on-board structural health monitoring system

Structural airframe maintenance is a subset of scheduled maintenance, and is performed at regular intervals to detect and repair cracks that would otherwise affect the safety of the airplane. It has been observed that only a fraction of airplanes undergo structural airframe maintenance at earlier scheduled maintenance times. But, intrusive inspection of all panels on the airplanes needs to be performed at the time of scheduled maintenance to ascertain the presence/absence of large cracks critical to the safety of the airplane. Recently, structural health monitoring techniques have been developed. They use on-board sensors and actuators to assess the current damage status of the airplane, and can be used as a tool to skip the structural airframe maintenance whenever deemed unnecessary. Two maintenance philosophies, scheduled structural health monitoring and condition-based maintenance skip, have been developed in this article to skip unnecessary structural airframe maintenances using the on-board structural health monitoring system. A cost model is developed to quantify the savings of these maintenance philosophies over scheduled maintenance.