Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Analytical modeling of rotor-structure coupling using modal decomposition for the structure and the blades.

This paper presents a linear semi-analytical model that is able to predict complex rotor-structure coupling phenomena and their stability. It was primarily designed so as to gain a better physical understanding of this kind of aeroelastic instabilities, triggering at higher frequencies than air and ground resonance, and involving several blade and structure modes. The analytical approach has a two-fold advantage since fast parametric studies can be carried out and a term-by-term analysis of the helicopter stability equations can be performed. In order to represent the elasticity of the structure and the blades, a modal decomposition method is introduced. The modal basis for the structure can either be obtained by a Finite Element Method or rigid degrees of freedom can be inputted. For the blades, a preliminary finite element routine is run, allowing for varying characteristics along the span. Blade offsets are introduced, and an unsteady aerodynamic model is implemented. The modal basis of the coupled system is then computed and a partial validation is done with HOST (Helicopter Overall Simulation Tool), a comprehensive aeroelastic code. Except for the built-in twist and the non-circulatory terms which are taken in a different manner in HOST and the presented model, the linearization results are similar. Future work using this model includes investigation of the helicopter stability thanks to parametric studies

Simulation of Heat and Mass Transport in a Square Lid-Driven Cavity with Proper Generalized Decomposition (PGD)

The aim of this study is to apply proper generalized decomposition (PGD) to solve mixed-convection problems with and without mass transport in a two dimensional lid-driven cavity. PGD is an iterative reduced order model approach which consists of solving a partial differential equation while seeking the solution in separated form. Comparisons with results in the literature and with results from a standard solver are make. For the case of a mixed-convection problem without mass transfer, three Richardson numbers are considered, Ri=0.1, Ri=1, and Ri=10. In this case, PGD is seven times faster than the standard solver with Ri=10 with a similar accuracy. For the case with mass transfer, simulations are done with different Lewis numbers, Le=5, Le=25, and Le=50, and with different value of the ratio N between the solutal and the thermal Grashoff numbers. In this case, too, PGD is ten times faster than the standard solver