An approach for eﬃcient, conceptual-level aerospace structural design using the static condensation reduced basis element method
Lee, Mario A.
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In order to improve the energy eﬃciency and environmental compliance of future aircraft, the aviation industry has sought to investigate the inclusion of a variety of new technologies that are capable of enabling these goals. Among these technologies is a suite of structural technologies that are aimed at reducing airframe weight. At the conceptual level of aircraft design, the issues of vehicle weight and technology impact are of paramount importance. In aerospace engineering literature, there is a consensus that the ﬁnite element method (FEM) is the most accurate numerical method for determining the structural behavior and consequently, the weight of structural concepts that do not have vast empirical weight data. In the areas of conceptual and preliminary level design, the ﬁnite element method is often used in tandem with numerical optimization techniques to enable design space exploration and for ﬁnding suitable candidates that meet the requirements for the design problem. Unfortunately, the inclusion of detailed ﬁnite element analysis into conceptual level design environments has traditionally been prohibitive because of the associated computational expense. Recently, there has been signiﬁcant interest in the development of reduced order modeling strategies that are capable of expediting analyses performed by high ﬁdelity simulations. Among these methods, a class of techniques known as Reduced Basis Approximation or Reduced Basis Methods has gained popularity because of their ability to replicate the accuracy of the higher ﬁdelity analyses but at a very small fraction of the computational cost. In particular, a recently proposed approach known as the “Static Condensation Reduced Basis Element (SCRBE) method” is quite attractive because of its versatility of modeling a wide variety of ﬁnal problem conﬁgurations with a relatively small data set. This approach has been demonstrated on large-scale problems with physical problem domains that can be constructed from a several repeated, underlying reference sub-domains or components. Unlike traditional reduced order modeling approaches, the SCRBE method performs the model reduction at the sub-domain level. This feature of the method enables the creation and analysis of a large variety of ﬁnal problem domain conﬁgurations that can all be modeled with underlying physics. The aim of this work is to develop an approach that uses the SCRBE method to enable conceptuallevel, linear-static, structural design/optimization. While there has been extensive development in the SCRBE method since its inception, the author was unable to ﬁnd many published, academic work that investigates the extension of this method to enable numerical optimization. Instead, most of the papers in literature focus on determining the state variable/ solution of the weak form of the underlying partial diﬀerential equation being modeled and then one or more outputs that depend on this solution. In the case of gradient-based optimization, one also needs the gradients of these outputs. For largescale problems, numerical diﬀerentiation is not viable due to the computational expense associated with the “curse-of-dimensionality.” This work presents an approach to estimate common, conceptual-level structural design metrics and their gradients under the SCRBE paradigm. Another observation from the literature is that there tends to be a disparity between the computational time required to compose the equations to be solved in the SCRBE method and the time required to actually solve these equations. The literature recommends certain operational procedures that can be taken advantage of to tackle this overhead. This includes the use of repeated, cloned sub-domains and interactive design. However, these methods may not be applicable during numerical optimization. Admittedly, certain implementation strategies (such as the use of parallel computation) can be used to help to alleviate this overhead. This thesis proposes a technique that addresses this computational overhead and is perhaps most beneﬁcial in situations where there are limited to moderate computational resources available. This technique leverages the matrix Discrete Empirical Interpolation Method (mDEIM). The developments in this thesis are illustrated on a simple canonical problem of the strength design of a membrane-loaded, patched, variable-stiﬀness, composite plate. The ﬁndings of the experiments indicate that the SCRBE method, plus the techniques that are added to address the eﬃciency of the method have the potential to enable eﬃcient conceptual-level structural design. It is anticipated that this approach can eventually be extended to conceptual-level studies of larger subsystems commonly featured in aerospace construction and forms an exciting avenue for future research.