Fractal Dimension of Dielectric Breakdown

Niemeyer, L.; Pietronero, L.; Wiesmann, H. J.

doi:10.1103/physrevlett.52.1033

Cited by 1,500 publications

(724 citation statements)

References 5 publications

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“…Another algorithm for generating the DLA-like cluster is the well-known DBM model [4]. Instead of tracing the motion of particles, one solves a Laplace equation and calculates a probability for a particle to hit the surface at the given site.…”

Section: Simulations a Dla Algorithmmentioning

confidence: 99%

“…The specific model which is a most common representation of this process is the diffusion limited aggregation (DLA) [3] and its generalization, dielectric breakdown model (DBM) [4] capturing most of the dynamic properties of the random aggregates evolution [1]. While analytical and numerical studies have had impressively advanced our understanding of the DLA [1,2], several critical issues remain unresolved.…”

Section: Introductionmentioning

confidence: 99%

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Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Menshutin

Shchur

Vinokour

2008

Physica A: Statistical Mechanics and its Applications

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A finite size effect in the probing of the harmonic measure in simulation of diffusion-limited aggregation (DLA) growth is investigated. We introduce variable size of the probe particles to estimate harmonic measure and extract fractal dimension of DLA clusters taking two limits, of vanishingly small probe particle size and of infinitely large size of a DLA cluster. We generate 1000 DLA clusters consisting of 50 million particles each using off-lattice killing-free algorithm developed in the early work. The introduced method leads to an unprecedented accuracy in the estimation of the fractal dimension. We discuss the variation of the probability distribution function with the size of probing particles.

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Section: Simulations a Dla Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Menshutin

Shchur

Vinokour

2008

Physica A: Statistical Mechanics and its Applications

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“…Although the physics behind the propagation of cracks, dielectric breakdowns, and flux avalanches, have been extensively studied in uniform media [2][3][4][5][6][7], little is known about the pattern formation in the case of periodic variations in the properties of the host material. In particular, one can then pose the question as to whether the morphology of flux avalanches is reflecting microscopic properties of the matrix or, inversely, if by introducing modulations on the material properties one would be able to impose a particular shape to the ubiquitous multi-branching splitting of avalanches.…”

Section: Introductionmentioning

confidence: 99%

Controllable morphology of flux avalanches in microstructured superconductors

et al. 2014

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The morphology of abrupt bursts of magnetic flux into superconducting films with engineered periodic pinning centers (antidots) has been investigated. Guided flux avalanches of thermomagnetic origin develop a tree-like structure, with the main trunk perpendicular to the borders of the sample, while secondary branches follow well-defined directions determined by the geometrical details of the underlying periodic pinning landscape. Strikingly, we demonstrate that in a superconductor with relatively weak random pinning, the morphology of such flux avalanches can be fully controlled by proper combinations of lattice symmetry and antidot geometry. Moreover, the resulting flux patterns can be reproduced, to the finest details, by simulations based on a phenomenological thermomagnetic model. In turn, this model can be used to predict such complex structures and to estimate physical variables of more difficult experimental access, such as the local values of temperature and electric field.

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“…The relative differences in the temperature values computed using Equation (13) are seen in Table 1 for the three horizon sizes and at different times. The plots in Figure 15 show the rates of δ-convergence for m = 6.5 and at times t = 0.05, 0.1, and 0.2 s. The rates are linear (1.09, 1.15, and 1.0, for the three horizon sizes, respectively).…”

Section: The Peridynamic Solution and Discussion Of Convergencementioning

confidence: 99%

“…Extraction of geothermal energy encounters fracturing rocks due to temperature gradients [11] and shrinkage due to cooling or drying causes cracks in basalt columns with specific spacings [12]. Dielectric breakdown of insulators [13][14][15], may also be modeled with the proposed peridynamic theory. Instead of having a conducting body with growing insulated cracks, in this case the body is an insulator in which conducting cracks grow and interact dynamically.…”

Section: Motivationmentioning

confidence: 99%

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

Bobaru

Duangpanya

2012

Journal of Computational Physics

249

142

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We introduce a multidimensional peridynamic formulation for transient heat-transfer. The model does not contain spatial derivatives and uses instead an integral over a region around a material point. By construction, the formulation converges to the classical heat transfer equations in the limit of the horizon (the nonlocal region around a point) going to zero. The new model, however, is suitable for modeling, for example, heat flow in bodies with evolving discontinuities such as growing insulated cracks. We introduce the peridynamic heat flux which exists even at sharp corners or when the isotherms are not smooth surfaces. The peridynamic heat flux coincides with the classical one in simple cases and, in general, it converges to it in the limit of the peridynamic horizon going to zero. We solve test problems and compare results with analytical solutions of the classical model or with other numerical solutions. Convergence to the classical solutions is seen in the limit of the horizon going to zero. We then solve the problem of transient heat flow in a plate in which insulated cracks grow and intersect thus changing the heat flow patterns. We also model heat transfer in a fiber-reinforced composite and observe transient but steep thermal gradients at the interfaces between the highly conductive fibers and the low conductivity matrix. Such thermal gradients can lead to delamination cracks in composites from thermal fatigue. The formulation may be used to, for example, evaluate effective thermal conductivities in bodies with an evolving distribution of insulating or permeable, possibly intersecting, cracks of arbitrary shapes.

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Fractal Dimension of Dielectric Breakdown

Cited by 1,500 publications

References 5 publications

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

Controllable morphology of flux avalanches in microstructured superconductors

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

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