From subspace learning to distance learning: A geometrical optimization approach

In this paper, we adopt a differential-geometry viewpoint to tackle the problem of learning a distance online. As this problem can be cast into the estimation of a fixed-rank positive semidefinite (PSD) matrix, we develop algorithms that exploits the rich geometry structure of the set of fixed-rank PSD matrices. We propose a method which separately updates the subspace of the matrix and its projection onto that subspace. A proper weighting of the two iterations enables to continuously interpolate between the problem of learning a subspace and learning a distance when the subspace is fixed. © 2009 IEEE.

A PDE viewpoint on basic properties of coordination algorithms with symmetries

Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform translation of all subsystem values. The present paper shows that in presence of such symmetry, fundamental properties can be highlighted by viewing the distributed system as the discrete approximation of a partial differential equation. An important fact is that the symmetry on the state space differs from the popular spatial invariance property, which is not necessary for the present results. The relevance of the viewpoint is illustrated on two examples: (i) ill-conditioning of interaction matrices in coordination/consensus problems and (ii) the string instability issue. ©2009 IEEE.

A perturbation approach to the stabilization of nonlinear cascade systems with time-delay

The effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semi-global stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback.

Geometric perspectives on MDO and MDO architectures

Two main perspectives have been developed within the Multidisciplinary Design Optimization (MDO) literature for classifying and comparing MDO architectures: a numerical point of view and a formulation/data flow point of view. Although significant work has been done here, these perspectives have not provided much in the way of a priori information or predictive power about architecture performance. In this report, we outline a new perspective, called the geometric perspective, which we believe will be able to provide such predictive power. Using tools from differential geometry, we take several prominent architectures and describe mathematically how each constructs the space through which it moves. We then consider how the architecture moves through the space which it has constructed. Taken together, these investigations show how each architecture relates to the original feasible design manifold, how the architectures relate to each other, and how each architecture deals with the design coupling inherent to the original system. This in turn lays the groundwork for further theoretical comparisons between and analyses of MDO architectures and their behaviour using tools and techniques derived from differential geometry. © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

A differential lyapunov framework for contraction analysis

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. © 2013 IEEE.

Optimization algorithms and ODE's in MDO

There is a need for a stronger theoretical understanding of Multidisciplinary Design Optimization (MDO) within the field. Having developed a differential geometry framework in response to this need, we consider how standard optimization algorithms can be modeled using systems of ordinary differential equations (ODEs) while also reviewing optimization algorithms which have been derived from ODE solution methods. We then use some of the framework's tools to show how our resultant systems of ODEs can be analyzed and their behaviour quantitatively evaluated. In doing so, we demonstrate the power and scope of our differential geometry framework, we provide new tools for analyzing MDO systems and their behaviour, and we suggest hitherto neglected optimization methods which may prove particularly useful within the MDO context. Copyright © 2013 by ASME.

Unified form language: A domain-specific language for weak formulations of partial differential equations

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies formultifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted. © 2014 ACM.

Differential equation based length scales to improve DES and RANS simulations

Surprisingly expensive to compute wall distances are still used in a range of key turbulence and peripheral physics models. Potentially economical, accuracy improving differential equation based distance algorithms are considered. These involve elliptic Poisson and hyperbolic natured Eikonal equation approaches. Numerical issues relating to non-orthogonal curvilinear grid solution of the latter are addressed. Eikonal extension to a Hamilton-Jacobi (HJ) equation is discussed. Use of this extension to improve turbulence model accuracy and, along with the Eikonal, enhance Detached Eddy Simulation (DES) techniques is considered. Application of the distance approaches is studied for various geometries. These include a plane channel flow with a wire at the centre, a wing-flap system, a jet with co-flow and a supersonic double-delta configuration. Although less accurate than the Eikonal, Poisson method based flow solutions are extremely close to those using a search procedure. For a moving grid case the Poisson method is found especially efficient. Results show the Eikonal equation can be solved on highly stretched, non-orthogonal, curvilinear grids. A key accuracy aspect is that metrics must be upwinded in the propagating front direction. The HJ equation is found to have qualitative turbulence model improving properties. © 2003 by P. G. Tucker.

Numerical analysis of the settlement induced damage to Palazzo Loggia in Brescia

One of the main causes of failure of historic buildings is represented by the differential settlements of foundations. Finite element analysis provides a useful tool for predicting the consequences of given ground displacements in terms of structural damage and also assesses the need of strengthening techniques. The actual damage classification for buildings subject to settlement bases the assessment of the potential damage on the expected crack pattern of the structure. In this paper, the correlation between the physical description of the damage in terms of crack width and the interpretation of the finite element analysis output is analyzed. Different discrete and continuum crack models are applied to simulate an experiment carried on a scale model of a masonry historical building, the Loggia Palace in Brescia (Italy). Results are discussed and a modified version of the fixed total strain smeared crack model is evaluated, in order to solve the problem related to the calculation of the exact crack width.

Assessment of the settlement vulnerability of masonry buildings

Excavation works in urban areas require a preliminary risk damage assessment. In historical cities, the prediction of building response to settlements is necessary to reduce the risk of damage of the architectural heritage. The current method used to predict the building damage due to ground deformations is the Limiting Tensile Strain Method (LTSM). In this approach the building is modelled as an elastic beam subjected to imposed Greenfield settlements and the induced tensile strains are compared with a limit value for the material. These assumptions can lead to a non realistic evaluation of the damage. In this paper, the possibility to apply a settlement risk assessment derived from the seismic vulnerability approach is considered. The parameters that influence the structural response to settlements can be defined through numerical analyses which take into account the nonlinear behaviour of masonry and the soil-structure interaction. The effects of factors like material quality, geometry of the structure, amount of openings, type of foundation or the actual state of preservation can be included in a global vulnerability index, which should indicate the building susceptibility to damage by differential settlements of a given magnitude. Vulnerability curves will represent the expected damage of each vulnerability class of building as a function of the settlement.