Guido Ridolfi

Robust Multi-Disciplinary Optimization of Unmanned Entry Capsules

The conceptual design of entry vehicles is commonly done in a number of sequential steps. One usually begins with a generic shape to get a first estimate of the aerodynamic properties and uses a mass-point model for the initial trajectory design. Gradually, more detail is added and the outer shape is changed to accommodate specific mission and/or trajectory requirements. This shape will largely define the aerothermodynamic characteristics of the vehicle. Since aerothermodynamic challenges, such as vehicle heating, remain one of the most difficult problems in atmospheric re-entry, an exploration of the possible shapes for a vehicle early in the design is advisable. It is advantageous to use a continuous model for the analysis, so that one is not limited to the analysis and comparison of a limited number of shapes, but is instead free to analyze any shape in the design space. With only 5 geometric parameters it is possible to already model the geometry of Apollo-like shapes, as demonstrated later in Section IV. The internal layout of the subsystems is usually addressed only at a later stage and the designers have to make sure that the mass properties (total mass, location of the center of mass and inertia tensor) meet the requirements. Deviations from these requirements can jeopardize the entire mission, because the loads on the vehicle may change, or the stability and control properties cannot be handled by the Guidance, Navigation, and Control (GNC) system any more. Further, uncertainties related to the entry conditions, environment, the characteristics of the thermal protection system, and the design characteristics and allocation of the equipments on board, pose the multidisciplinary problem to be particularly cumbersome. In this paper we propose a multidisciplinary, robust optimization approach for the design of unmanned entry capsules in support of the activities of the International Space Station. This problem is handled by minimizing the total mass of the capsules, while maximizing the internal available volume for carrying payload. As a third objective, we propose the maximization of the re-usability of the capsules, which can be seen as an attempt to push towards cheaper and more efficient solutions. The shape, aerothermodynamic, and dynamic mathematical models are adapted from the work of Dirkx and Mooij. It was demonstrated that the proposed simplified aerodynamic model can predict the aerodynamic forces and moments for ballistic shapes sufficiently well for use at a conceptual design stage. The multidisciplinary design framework is now enriched with a Thermal-Protection System (TPS) model, encompassing re-usable and ablative materials, as well as active cooling mechanisms. This allows for a complete conceptual design of an entry capsule. Uncertainties of the design variables and environmental factors are integrated into the optimization process to handle probabilistic constraints. A probabilistic constraint is a constraint in the design or objective space that shall be satisfied with a pre-defined confidence level. The optimizer thus drives the search of optimal capsules towards those solutions that have the best expected performance under uncertain conditions, and that also meet the constraints with a given confidence level, pre-selected by the designer/decision-maker. A sampling-based approach is used to estimate the expected performance of the capsules and to determine the compliance with the probabilistic constraints. For each design point to be evaluated by the optimizer, a set of additional design points is generated around it, according to the joint Probability Density Function (PDF) of the uncertain variables and uncertain environmental factors, and evaluated. To limit the computational effort of the robust optimization, we adopt a double-repository archive maintenance scheme to save all the design-variable combinations computed during the process such that previous design points can be reused at future steps. The double-repository scheme allows to preserve the joint PDF of the input uncertain variables, therefore it is generally applicable with any type of multivariate distribution as input. This paper is structured as follows. In Section II we provide a brief overview of related work. We then introduce the robust optimization approach and the double-repository archive maintenance scheme in Section III. A short overview of the mathematical models used for the analysis can be found in Section IV. In Section V results are discussed and in Section VI we provide conclusions and recommendations. Two appendix sections, namely Sections VII and VIII provide the thermophysical properties of the materials used for the TPS concepts and the results of the validation of the thermal models respectively.

A Modeling Framework for the Concurrent Design of Complex Space Systems

The design of complex systems has become more and more articulated during the last decade, thus forcing radical modifications on the overall methodological approach. The authors developed a design methodology, which allows the user to design a particular category of complex systems usually called System-of-systems. This paper discusses the general framework to deal with the decomposition of a system-of-systems in its elements and sub-elements, to enable a faster and more effective solution of the problem, extending the applicability of the Concurrent Engineering paradigm to design phases that go beyond the preliminary/conceptual one. A hypothetical space exploration architecture, with a rover system, a lander system and an Earth-Moon transfer mission, have been implemented and discussed, linking the elements of this particular System-of-systems with a non-hierarchical decomposition approach and a multi-disciplinary feasible formulation.

Regression-Based Sensitivity Analysis and Robust Design

This paper presents the Regression-Based global Sensitivity Analysis method (RBSA). It is an approach for quantitative, variance-based, sensitivity analysis of mathematical models used for design purposes. The method is based on the subdivision of the global variance into its components, due to the design-factor contributions, using general polynomial regression models. The performance of the RBSA is compared to other methods commonly used in engineering for computing sensitivity, namely, the method of Sobol’, the Fourier amplitude sensitivity test, the method of Morris, and the standardized regression coefficients. It was found that RBSA, under certain circumstances, provides very accurate results with a significant reduction of the number of required model evaluations. A test case, using the mathematical models of two subsystems of a spacecraft, demonstrates how RBSA facilitates the discovery and understanding of the effects of the design choices on the performance of the system.