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In this work a minimization problem for the magnetic Ginzburg–Landau functional in a circular domain with randomly distributed small holes is considered. We develop a new analytical approach for solving a non-standard boundary value problem for the magnetic field presented as a function of the |$n$|-tuple of degrees of vortices pinned by the |$n$|...
Contexts in source publication
Context 1
... will be shown later that this restriction is imposed so that the method developed will converge using successive approximations. Similar to the approach in Beryland & Rybalko (2013), we consider the vortex distribution (vorticity) on the perforated domain D with a large, but finite number n ε of pinning sites (holes, see Fig. 1) where T k represents the boundary of these pinning sites. In order to find this distribution one must find the minimizers of the non-dimensional GL energy ...
Context 2
... a sufficiently large number of walks (≈ 80) the obtained location of the centre is considered a statistical realization of the random distribution U (Czapla et al., 2012a,b). The typical value for ρ used in the simulation results presented is ρ := 1 5 ( ω 1 √ n − 2r), where ω 1 is the domain size and n is the number of holes. ...
Context 3
... will be shown later that this restriction is imposed so that the method developed will converge using successive approximations. Similar to the approach in Beryland & Rybalko (2013), we consider the vortex distribution (vorticity) on the perforated domain D with a large, but finite number n ε of pinning sites (holes, see Fig. 1) where T k represents the boundary of these pinning sites. In order to find this distribution one must find the minimizers of the non-dimensional GL energy ...
Context 4
... a sufficiently large number of walks (≈ 80) the obtained location of the centre is considered a statistical realization of the random distribution U (Czapla et al., 2012a,b). The typical value for ρ used in the simulation results presented is ρ := 1 5 ( ω 1 √ n − 2r), where ω 1 is the domain size and n is the number of holes. ...
Citations
Many attempts were undertaken to modify Maxwell’s approach in the theory of composites. Self-consistent methods (effective medium approximation, mean field, Mori-Tanaka methods, reiterated homogenization etc) were advanced to determine the effective properties of composites. It is demonstrated by an example that these extensions are methodologically misleading. They lead to a plenty of illusory different formulas reduced to the Maxwell type, lower order estimation for dilute composites.
The local fields in composites and porous media can have complicated structure because of the fine geometrical inhomogeneity. Numeric examples show that randomly generated local fields can have complicated fractal structure contrary to the local fields in periodic composites. It is demonstrated that the local oscillations of the stresses in random composites are higher than in regular ones. This result implies that the damage risk is higher for irregular elastic composites than for regular ones ceteris paribus. For viscous fluid, this means that irregular locations of obstacles increase local oscillations of the velocity, hence, lead to turbulence. We consider fields governed by the 2D Poisson equation in a perforated domain corresponding to the host material. The holes of correspond to the torsion problem in elasticity and the hard disks to longitudinal flow of viscous fluid. The corresponding Dirichlet problem in randomly generated multiply connected domains is solved. A method of functional equations is applied following Mityushev’s scheme for the Riemann–Hilbert type problems in multiply connected domains. It is justified that the rate of convergence for functional equations is of order where is the connectivity of the domain whose linear size is normalized to unity and the radii of holes. This observation shows that for large and sufficiently small few iterations for the functional equations can give an acceptable numerical result.
