267 research outputs found
On the dynamics of crystalline motions
Solids can exist in polygonal shapes with boundaries unions of flat -pieces· called· facets. Analyzing the- growth -of such crystalline shapes is an important problem in materials science. In this paper we derive equations that govern the evolution of such shapes; we formulate the corresponding initial-value problem variationally; and we use this formulation to establish a comparison principle for crystalline evolutions. This principle asserts that two evolving crystals one initially inside the other will remain in that configuration for all time
Thermodynamics of non-local materials: extra fluxes and internal powers
The most usual formulation of the Laws of Thermodynamics turns out to be
suitable for local or simple materials, while for non-local systems there are
two different ways: either modify this usual formulation by introducing
suitable extra fluxes or express the Laws of Thermodynamics in terms of
internal powers directly, as we propose in this paper. The first choice is
subject to the criticism that the vector fluxes must be introduced a posteriori
in order to obtain the compatibility with the Laws of Thermodynamics. On the
contrary, the formulation in terms of internal powers is more general, because
it is a priori defined on the basis of the constitutive equations. Besides it
allows to highlight, without ambiguity, the contribution of the internal powers
in the variation of the thermodynamic potentials. Finally, in this paper, we
consider some examples of non-local materials and derive the proper expressions
of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic
A Gauge field Induced by the Global Gauge Invariance of Action Integral
As a general rule, it is considered that the global gauge invariance of an
action integral does not cause the occurrence of gauge field. However, in this
paper we demonstrate that when the so-called localized assumption is excluded,
the gauge field will be induced by the global gauge invariance of the action
integral. An example is given to support this conclusion.Comment: 13 pages. Some typing errors are corrected and the format is update
Periodic Homogenization and Material Symmetry in Linear Elasticity
Here homogenization theory is used to establish a connection between the
symmetries of a periodic elastic structure associated with the microscopic
properties of an elastic material and the material symmetries of the effective,
macroscopic elasticity tensor. Previous results of this type exist but here
more general symmetries on the microscale are considered. Using an explicit
example, we show that it is possible for a material to be fully anisotropic on
the microscale and yet the symmetry group on the macroscale can contain
elements other than plus or minus the identity. Another example demon- strates
that not all material symmetries of the macroscopic elastic tensor are
generated by symmetries of the periodic elastic structure.Comment: 18 pages, 5 figure
A second order minimality condition for the Mumford-Shah functional
A new necessary minimality condition for the Mumford-Shah functional is
derived by means of second order variations. It is expressed in terms of a sign
condition for a nonlocal quadratic form on , being a
submanifold of the regular part of the discontinuity set of the critical point.
Two equivalent formulations are provided: one in terms of the first eigenvalue
of a suitable compact operator, the other involving a sort of nonlocal capacity
of . A sufficient condition for minimality is also deduced. Finally, an
explicit example is discussed, where a complete characterization of the domains
where the second variation is nonnegative can be given.Comment: 30 page
Three-points interfacial quadrature for geometrical source terms on nonuniform grids
International audienceThis paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells' size, for which -error estimates, , are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem's data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics)
Size-structured populations: immigration, (bi)stability and the net growth rate
We consider a class of physiologically structured population models, a first
order nonlinear partial differential equation equipped with a nonlocal boundary
condition, with a constant external inflow of individuals. We prove that the
linearised system is governed by a quasicontraction semigroup. We also
establish that linear stability of equilibrium solutions is governed by a
generalized net reproduction function. In a special case of the model
ingredients we discuss the nonlinear dynamics of the system when the spectral
bound of the linearised operator equals zero, i.e. when linearisation does not
decide stability. This allows us to demonstrate, through a concrete example,
how immigration might be beneficial to the population. In particular, we show
that from a nonlinearly unstable positive equilibrium a linearly stable and
unstable pair of equilibria bifurcates. In fact, the linearised system exhibits
bistability, for a certain range of values of the external inflow, induced
potentially by All\'{e}e-effect.Comment: to appear in Journal of Applied Mathematics and Computin
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