有哪些信誉好的足球投注网站Equilibrium and absolute minimal states of Mumford-Shah functionals and brittle fracture pr.pdf
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Equilibrium and absolute minimal states of Mumford-Shah functionals and brittle fracture pr
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Equilibrium and absolute minimal states of Mumford-Shah
functionals and brittle fracture propagation
Marius Buliga?
Abstract
By a combination of geometrical and configurational analysis we study the
properties of absolute minimal and equilibrium states of general Mumford-Shah
functionals, with applications to models of quasistatic brittle fracture propagation.
The main results concern the mathematical relations between physical quantities
as energy release rate and energy concentration for 3D cracks with complex shapes,
seen as outer measures living on the crack edge.
Keywords: 3D brittle fracture; energy methods; Mumford-Shah functional
1 Introduction
A new direction of research in brittle fracture mechanics begins with the article of
Mumford Shah [12] regarding the problem of image segmentation. This problem,
which consists in finding the set of edges of a picture and constructing a smoothed
version of that picture, it turns to be intimately related to the problem of brittle crack
evolution. In the before mentioned article Mumford and Shah propose the following
variational approach to the problem of image segmentation: let g : ? ? R2 → [0, 1] be
the original picture, given as a distribution of grey levels (1 is white and 0 is black), let
u : ? → R be the smoothed picture and K be the set of edges. K represents the set
where u has jumps, i.e. u ∈ C1(? \K,R). The pair formed by the smoothed picture u
and the set of edges K minimizes then the functional:
I(u,K) =
∫
?
α | ?u |2 dx +
∫
?
β | u? g |2 dx + γH1(K) .
The parameter α controls the smoothness of the new picture u, β controls the L2
distance between the smoothed picture and the original one and γ controls the total
length of the edges given by this variational method. The authors remark that for β = 0
the functional I might be useful for an energetic treatment of fracture mechanics.
An energetic approach to fracture mechanics is natur
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