Study of the model-order reduction of the aerolastic behavior of a wing

The  ultimate goal of this project is to construct a   reduced-order model capable of  providing real-time predictions of   the aeroelastic behavior of a wing.  The approach for carrying out such a task is, firstly, in the spirit of classic modal analysis, to project the full-order, governing equations of the wing (finite element equations, for instance) onto the low-dimensional subspace spanned by a few global displacement modes. Such displacement modes, in turn, are obtained by applying data compression algorithms  to a representative set of full-order simulations.   Once these dominant displacement modes have been identified, the next step in the approach is to choose, among all points of the underlying finite element mesh,  a set of sampling points  so that the integrals appearing in the weak form of the balance equation can be accurately evaluated by monitoring the strains and stresses only at such key points.The main objective of this paper is to apply the model-order reduction technique to an airplane’s wing in order to speed up development of aircrafts or to get real-time results of a plane structural state. However, this case is especially complex since the wings are an aeroelastic problem where both fluid and structure must be computed in order to get realistic results. In order to improve the overall airplane design speed -in addition to the usage of MOR techniques- a complementary software has been developed. This is a parametric software capable of quickly generating a geometry and exporting it to simulate both the fluid and the structure with a FE software like Kratos. This software will be open sourced. The usage of the custom software helps to generate geometries that differ only on a single design parameter (the angle of attack in this paper). These different geometries are then processed with Kratos to obtain the high-fidelity result from each one of them. Once the high-fidelity snapshots have been obtained (five are used in this paper), the reduced order models are generated using a discrete version of the Proper Orthogonal Decomposition (POD) called Single Value Decomposition (SVD). Finally, using the discrete empirical interpolation method (DEIM), it is possible to interpolate between the simulations and obtain the results of any intermediate state in less than a second without having to perform the full simulation. No physical model has been constructed to compute the fluid and only statistical methods are employed for that part. The results turned out to be very precise regarding the structure ROM; all the same, the only statistical approach to the fluid proved to be not ideal and the accuracy error remained around 15% for this part

Some word order biases from limited brain resources: A mathematical approach

In this paper, we propose a mathematical framework for studying word order optimization. The framework relies on the well-known positive correlation between cognitive cost and the Euclidean distance between the elements (e.g. words) involved in a syntactic link. We study the conditions under which a certain word order is more economical than an alternative word order by proposing a mathematical approach. We apply our methodology to two different cases: (a) the ordering of subject (S), verb (V) and object (O), and (b) the covering of a root word by a syntactic link. For the former, we find that SVO and its symmetric, OVS, are more economical than OVS, SOV, VOS and VSO at least 2/3 of the time. For the latter, we find that uncovering the root word is more economical than covering it at least 1/2 of the time. With the help of our framework, one can explain some Greenbergian universals. Our findings provide further theoretical support for the hypothesis that the limited resources of the brain introduce biases toward certain word orders. Our theoretical findings could inspire or illuminate future psycholinguistics or corpus linguistics studies.Peer ReviewedPostprint (author's final draft

Why do syntactic links not cross?

Here we study the arrangement of vertices of trees in a 1-dimensional Euclidean space when the Euclidean distance between linked vertices is minimized. We conclude that links are unlikely to cross when drawn over the vertex sequence. This finding suggests that the uncommonness of crossings in the trees specifying the syntactic structure of sentences could be a side-effect of minimizing the Euclidean distance between syntactically related words. As far as we know, nobody has provided a successful explanation of such a surprisingly universal feature of languages that was discovered in the 60s of the past century by Hays and Lecerf. On the one hand, support for the role of distance minimization in avoiding edge crossings comes from statistical studies showing that the Euclidean distance between syntactically linked words of real sentences is minimized or constrained to a small value. On the other hand, that distance is considered a measure of the cost of syntactic relationships in various frameworks. By cost, we mean the amount of computational resources needed by the brain. The absence of crossings in syntactic trees may be universal just because all human brains have limited resources.Peer ReviewedPostprint (author's final draft

A short survey on observability

The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in \emph{geometric invariant theory}. The concept of \emph{observable subgroup} was introduced in the early 1960s with the purpose of studying extensions of representations from an affine algebraic subgroup to the whole group. The extent of its importance in \emph{representation and invariant theory} in particular for Hilbert's $14^{\text{th}}$ problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of \emph{strong observability} was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of \emph{observable action} of an affine algebraic group on an affine variety, launching a series of new applications. In 2006 the related concept of \emph{observable adjunction} was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned