3,011 research outputs found
Currents and finite elements as tools for shape space
The nonlinear spaces of shapes (unparameterized immersed curves or
submanifolds) are of interest for many applications in image analysis, such as
the identification of shapes that are similar modulo the action of some group.
In this paper we study a general representation of shapes that is based on
linear spaces and is suitable for numerical discretization, being robust to
noise. We develop the theory of currents for shape spaces by considering both
the analytic and numerical aspects of the problem. In particular, we study the
analytical properties of the current map and the norm that it induces
on shapes. We determine the conditions under which the current determines the
shape. We then provide a finite element discretization of the currents that is
a practical computational tool for shapes. Finally, we demonstrate this
approach on a variety of examples
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
Void Formation and Roughening in Slow Fracture
Slow crack propagation in ductile, and in certain brittle materials, appears
to take place via the nucleation of voids ahead of the crack tip due to plastic
yields, followed by the coalescence of these voids. Post mortem analysis of the
resulting fracture surfaces of ductile and brittle materials on the m-mm
and the nm scales respectively, reveals self-affine cracks with anomalous
scaling exponent in 3-dimensions and in
2-dimensions. In this paper we present an analytic theory based on the method
of iterated conformal maps aimed at modelling the void formation and the
fracture growth, culminating in estimates of the roughening exponents in
2-dimensions. In the simplest realization of the model we allow one void ahead
of the crack, and address the robustness of the roughening exponent. Next we
develop the theory further, to include two voids ahead of the crack. This
development necessitates generalizing the method of iterated conformal maps to
include doubly connected regions (maps from the annulus rather than the unit
circle). While mathematically and numerically feasible, we find that the
employment of the stress field as computed from elasticity theory becomes
questionable when more than one void is explicitly inserted into the material.
Thus further progress in this line of research calls for improved treatment of
the plastic dynamics.Comment: 15 pages, 20 figure
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