Self-similar Model Research Articles

Characterizing the acceleration time of laser-driven ion acceleration with data-informed neural networks

Peak ion energy is an important figure-of-merit in short-pulse, laser-driven ion acceleration and is dependent on an associated acceleration time. Standard metrics for these quantities depend on analytical results such as the self-similar fluid model or empirical models based on relatively small experimental and simulation datasets. In this work we attempt to use a data-informed neural network (NN) as a surrogate model for a large ensemble of PIC simulations to investigate an effective acceleration time. We explore the application of a stacked convolutional and recurrent NN architecture for improved regression by incorporating the time dependencies of the data into the training process. Of particular note is how pretraining a network on lower fidelity data, e.g. 1D analytical results, greatly improves the network’s ability to learn more complex, higher fidelity data. Finally, the dependency of the acceleration time on various laser and plasma parameters is explored.

The Diffusion Coefficient of the Splashback Mass Function as a Probe of Cosmology

Abstract We present an analytic model for the splashback mass function of dark matter halos, which is parameterized by a single coefficient and constructed in the framework of the generalized excursion set theory and the self-similar spherical infall model. The value of the single coefficient that quantifies the diffusive nature of the splashback boundary is determined at various redshifts by comparing the model with the numerical results from the Erebos N-body simulations for the Planck and the WMAP7 cosmologies. Showing that the analytic model with the best-fit coefficient provides excellent matches to the numerical results in the mass range of 5 ≤ M/(1012 h −1 M ⊙) < 103, we employ the Bayesian and Akaike Information Criterion tests to confirm that our model is most preferred by the numerical results compared to previous models at redshifts of 0.3 ≤ z ≤ 3 for both of the cosmologies. We also found that the diffusion coefficient decreases almost linearly with redshift, converging to zero at a certain threshold redshift, z c , whose value significantly differs between the Planck and WMAP7 cosmologies. Our result implies that the splashback mass function of dark matter halos at z ≥ z c is well described by a parameter-free analytic formula and that z c may have the potential to independently constrain the initial conditions of the universe.

AN ANALYTICAL MODEL FOR PORE AND TORTUOSITY FRACTAL DIMENSIONS OF POROUS MEDIA

Accurate characterization of pore-scale structures of porous media is necessary for studying their transport mechanisms and properties. An analytical model for pore and capillary structures of porous media is developed based on fractal theory in this study. The pore and tortuosity fractal dimensions are introduced to characterize the pore size distribution and tortuous flow paths. A power law scaling between fractal probability function and pore diameter is proposed, which can be applied to determine the pore fractal dimension. The explicit expression for tortuosity fractal dimension is derived based on exactly self-similar fractal set and fractal capillary bundle model. The present fractal model has been validated by comparison with that of experiments and numerical simulations as well as theoretical models. The results show that the tortuosity fractal dimension decreases as porosity and pore fractal dimension increase, it increases with the increment of tortuosity. Both the particle shape and pore size range take important effect on the tortuosity fractal dimension under certain porosity. The proposed pore-scale model can present a conceptual tool to study the transport mechanisms of porous media and may provide useful guideline for oil and gas exploitation, hydraulic resource development, geotechnical engineering and chemical engineering.

Acoustics of Fractal Porous Material and Fractional Calculus

In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties.

Dynamical equilibrium states of a class of irrotational non-orthogonally transitive G 2 cosmologies: II. Models with one hypersurface-orthogonal Killing vector field

We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. It is assumed throughout that the source is a perfect fluid with linear equation of state. We restrict the similarity group to possess both an Abelian G 2, and a single hypersurface orthogonal Killing vector field, and we restrict the fluid flow to be orthogonal to the orbits of the Abelian G 2. The temporal evolution of the models is forced to be power law, due to the similarity group, and the Einstein field equations reduce to a three-dimensional autonomous system of ordinary differential equations which is qualitatively analysed in order to determine the spatial structure of the models. The existence of two classes of well-behaved models is demonstrated. The first of these is asymptotically spatially homogeneous and matter dominated, and the second is vacuum dominated and either asymptotically spatially homogeneous or acceleration dominated, at large spatial distances.

The general applicability of self-similar solutions for thermal disc winds

ABSTRACT Thermal disc winds occur in many contexts and may be particularly important to the secular evolution and dispersal of protoplanetary discs heated by high energy radiation from their central star. In this paper, we generalize previous models of self-similar thermal winds – which have self-consistent morphology and variation of flow variables – to the case of launch from an elevated base and to non-isothermal conditions. These solutions are well-reproduced by hydrodynamic simulations, in which, as in the case of isothermal winds launched from the midplane, we find winds launch at the maximum Mach number for which the streamline solutions extend to infinity without encountering a singularity. We explain this behaviour based on the fact that lower Mach number solutions do not fill the spatial domain. We also show that hydrodynamic simulations reflect the corresponding self-similar models across a range of conditions appropriate to photoevaporating protoplanetary discs, even when gravity, centrifugal forces, or changes in the density gradient mean the problem is not inherently scale free. Of all the parameters varied, the elevation of the wind base affected the launch velocity and flow morphology most strongly, with temperature gradients causing only minor differences. We explore how launching from an elevated base affects Ne ii line profiles from winds, finding it increases (reduces) the full width at half maximum (FWHM) of the line at low (high) inclination to the line of sight compared with models launched from the disc midplane and thus weakens the dependence of the FWHM on inclination.

Erratum: A self-similar dynamic rupture model based on the simplified wave-rupture analogy

Erratum: A self-similar dynamic rupture model based on the simplified wave-rupture analogy

On the Possibility of Describing the Microstructure of Human Dentin as Soft Matrix Filled by Solid Particles

Experimental verification of the applicability of self-similar hierarchical mechanics model for calculation of the mechanical properties of human dentin was carried out. A comparison of the deformation behavior of human dentin with the several polymer based dental composites under compression and bending was performed. It was shown that the deformation behavior of some highly filled composites was close to human dentin in the elastic regime whereas they cannot be deformed plastically under compression. Low filled composites possess higher plasticity, but a lower Young’s modulus than dentin under compression. Mechanical properties of a highly filled composite (∼60%, by volume) with smaller size of fillers are closer to the properties of human dentin under bending whereas other tested dental materials have a lower Young’s modulus and flexural strength in comparison with dentin. Therefore, we concluded that it was impossible to achieve complete combination of the mechanical properties of human dentin such as high elasticity and plasticity at high strength by means of varying the amount and size of solid filler in polymer matrix. Self-similar hierarchical mechanics model can be used for describing of the deformation behavior of human dentin in elastic regime and one level is sufficient for these calculations.

Evolution of splashback boundaries and gaseous outskirts: insights from mergers of self-similar galaxy clusters

ABSTRACT A self-similar spherical collapse model predicts a dark matter (DM) splashback and accretion shock in the outskirts of galaxy clusters while missing a key ingredient of structure formation – processes associated with mergers. To fill this gap, we perform simulations of merging self-similar clusters and investigate their DM and gas evolution in an idealized cosmological context. Our simulations show that the cluster rapidly contracts during the major merger and the splashback radius rsp decreases, approaching the virial radius rvir. While in the self-similar model rsp depends on a smooth mass accretion rate parameter Γs, our simulations show that in the presence of mergers, rsp responds to the changes in the total mass accretion rate Γvir, which accounts for both mergers and smooth accretion. The scatter of the Γvir − rsp/rvir relation indicates a generally low Γs ∼ 1 in clusters in cosmological simulations. In contrast to the DM, the hot gaseous atmospheres significantly expand by the merger-accelerated (MA-) shocks formed when the runaway merger shocks overtake the outer accretion shock. After a major merger, the MA-shock radius is larger than rsp by a factor of up to ∼1.7 for Γs ≲ 1 and is ∼rsp for Γs ≳ 3. This implies that (1) mergers could easily generate the MA-shock-splashback offset measured in cosmological simulations, and (2) the smooth mass accretion rate is small in regions away from filaments where MA-shocks reside. We further discuss the shapes of the DM haloes, various shocks, and contact discontinuities formed at different epochs of the merger, and the ram-pressure stripping in cluster outskirts.

Single depth map super-resolution via joint non-local self-similarity modeling and local multi-directional gradient-guided regularization

The development of consumer depth cameras makes it possible to acquire depth information of a scene in real-time. However, low resolution and low quality of a depth map has greatly constrained its applications. In this paper, we propose a novel framework for single depth map super-resolution, which considers local and non-local information jointly in the depth map. For the non-local constraint, group-based sparse representation is used to explore non-local self-similarity in the depth map. For the local constraint, a multi-directional gradient-guided regularization is proposed to describe the gradient of the depth map with spatially varying orientations. The former constraint contains the visual artifacts effectively, while the latter restores sharp edge and fine structure. Finally, the two complementary regularizers are jointly casted into a unified optimization framework, where a split Bregman-based technique is developed to tackle the optimization problem. Both quantitative and qualitative evaluations indicate that the proposed method can obtain better reconstruction performance compared with state-of-the-art methods.

Protoplanetary disk formation from the collapse of a prestellar core

Context.Between the two research communities that study star formation and protoplanetary disk evolution, only a few efforts have been made to understand and bridge the gap between studies of a collapsing prestellar core and a developed disk. While it has generally been accepted for about a decade that the magnetic field and its nonideal effects play important roles during the stellar formation, simple models of pure hydrodynamics and angular momentum conservation are still widely employed in the studies of disk assemblage in the framework of the so-called alpha-disk model because these models are simple.Aims.We revisit the assemblage phase of the protoplanetary disk and employ current knowledge of the prestellar core collapse.Methods.We performed 3D magnetohydrodynamic (MHD) simulations with ambipolar diffusion and full radiative transfer to follow the formation of the protoplanetary disk within a collapsing prestellar core. The global evolution of the disk and its internal properties were analyzed to understand how the infalling envelope regulates the buildup and evolution of the disk. We followed the global evolution of the protoplanetary disk from the prestellar core collapse during 100 kyr with a reasonable resolution of AU. Two snapshots from this reference run were extracted and rerun with significantly increased resolution to resolve the interior of the disk.Results.The disk that formed under our simulation setup is more realistic and agrees with recent observations of disks around class 0 young stellar objects. The source function of the mass flux that arrives at the disk and the radial mass accretion rate within the disk are measured and compared to analytical self-similar models based on angular momentum conservation. The source function is very centrally peaked compared to classical hydrodynamical models, implying that most of the mass falling onto the star does not transit through the midplane of the disk. We also found that the disk midplane is almost dead to turbulence, whereas upper layers and the disk outer edge are highly turbulent, and this is where the accretion occurs. The snow line, located at about 5–10 AU during the infall phase, is significantly farther away from the center than in a passive disk. This result might be of numerical origin.Conclusions.We studied self-consistent protoplanetary disk formation from prestellar core collapse, taking nonideal MHD effects into account. We developed a zoomed rerun technique to quickly obtain a reasonable disk that is highly stratified, weakly magnetized inside, and strongly magnetized outside. During the class 0 phase of protoplanetary disk formation, the interaction between the disk and the infalling envelope is important and ought not be neglected. We measured the complex flow pattern and compared it to the classical models of pure hydrodynamical infall. Accretion onto the star is found to mostly depend on dynamics at large scales, that is, the collapsing envelope, and not on the details of the disk structure.

Correct proof of the main result in “The number of spanning trees of a class of self-similar fractal models” by Ma and Yao

The problem of counting the number of spanning trees of a network built by a replacement procedure that yields a self-similar structure is considered. This problem has been receiving growing attention in the specialized literature in the recent years. One of the important measures of the global reliability of a network is the number of spanning trees. In this paper, we present a correction of the two main theorems claimed by Ma and Yao [7] concerning the number and the entropy of spanning trees of a class of self-similar fractal models with proofs of the new results. Also, the numerical values of the number of spanning trees are obtained and compared with the values of the formula in Theorem 3.1 of [7].

A self-similar dynamic rupture model based on the simplified wave-rupture analogy

SUMMARYThe investigation of stresses, faults, structure and seismic hazards requires a good understanding and mapping of earthquake rupture and slip. Constraining the finite source of earthquakes from seismic and geodetic waveforms is challenging because the directional effects of the rupture itself are small and dynamic numerical solutions often include a large number of free parameters. The computational effort is large and therefore difficult to use in an exploratory forward modelling or inversion approach. Here, we use a simplified self-similar fracture model with only a few parameters, where the propagation of the fracture front is decoupled from the calculation of the slip. The approximative method is flexible and computationally efficient. We discuss the strengths and limitations of the model with real-case examples of well-studied earthquakes. These include the Mw 8.3 2015 Illapel, Chile, megathrust earthquake at the plate interface of a subduction zone and examples of continental intraplate strike-slip earthquakes like the Mw 7.1 2016 Kumamoto, Japan, multisegment variable slip event or the Mw 7.5 2018 Palu, Indonesia, supershear earthquake. Despite the simplicity of the model, a large number of observational features ranging from different rupture-front isochrones and slip distributions to directional waveform effects or high slip patches are easy to model. The temporal evolution of slip rate and rise time are derived from the incremental growth of the rupture and the stress drop without imposing other constraints. The new model is fast and implemented in the open-source Python seismology toolbox Pyrocko, ready to study the physics of rupture and to be used in finite source inversions.

Evolution of the Thermodynamic Properties of Clusters of Galaxies out to Redshift of 1.8

Abstract The thermodynamic properties of the hot plasma in galaxy clusters retain information on the processes leading to the formation and evolution of the gas in their deep, dark matter potential wells. These processes are dictated not only by gravity but also by gas physics, e.g., active galactic nucleus feedback and turbulence. In this work, we study the thermodynamic properties, e.g., density, temperature, pressure, and entropy, of the most massive and the most distant (seven clusters at z > 1.2) clusters selected by the South Pole Telescope and compare them with those of the nearby clusters (13 clusters at z < 0.1) to constrain their evolution as a function of time and radius. We find that thermodynamic properties in the outskirts of high-redshift clusters are remarkably similar to the low-redshift clusters, and their evolution follows the prediction of the self-similar model. Their intrinsic scatter is larger, indicating that the physical properties that lead to the formation and virialization of cluster outskirts show evolving variance. On the other hand, thermodynamic properties in the cluster cores deviate significantly from self-similarity, indicating that the processes that regulate the core are already in place in these very high redshift clusters. This result is supported by the unevolving physical scatter of all thermodynamic quantities in cluster cores.

Effects of gain distribution on self-similar amplification of picosecond pulses

The effects of gain distribution on self-similar amplification of picosecond pulses in a Yb-doped fiber laser system were studied by numerical simulation. Ultrashort laser pulses amplified in self-similar amplification theoretical model was established to analyze the impact of pump configuration, fiber length and total gain coefficient on the self-similar amplification evolution and laser output performance. Detailed numerical simulation reveals that the best self-similar amplification result can be found for different cases, where high-quality self-similar pulses with ~100 fs transform-limited pulse duration are obtained. It is demonstrated that the self-similar evolution speed in a forward-pumping scheme is faster than that in a backward-pumping scheme for a fixed seed pulse. Furthermore, the results indicate that for the self-similar amplifier with different fiber lengths and gain coefficients, the forward-pumping scheme shows better evolution results in lower seed energy and longer wavelength range, while the backward-pumping scheme is more suitable for the higher seed energy and shorter wavelength range.

Heavy-Tailed Self-Similarity Modeling for Single Image Super Resolution.

Self-similarity is a prominent characteristic of natural images that can play a major role when it comes to their denoising, restoration or compression. In this paper, we propose a novel probabilistic model that is based on the concept of image patch similarity and applied to the problem of Single Image Super Resolution. Based on this model, we derive a Variational Bayes algorithm, which super resolves low-resolution images, where the assumed distribution for the quantified similarity between two image patches is heavy-tailed. Moreover, we prove mathematically that the proposed algorithm is both an extended and superior version of the probabilistic Non-Local Means (NLM). Its prime advantage remains though, which is that it requires no training. A comparison of the proposed approach with state-of-the-art methods, using various quantitative metrics shows that it is almost on par, for images depicting rural themes and in terms of the Structural Similarity Index (SSIM) with the best performing methods that rely on trained deep learning models. On the other hand, it is clearly inferior to them, for urban themed images and in terms of all metrics, especially for the Mean-Squared-Error (MSE). In addition, qualitative evaluation of the proposed approach is performed using the Perceptual Index metric, which has been introduced to better mimic the human perception of the image quality. This evaluation favors our approach when compared to the best performing method that requires no training, even if they perform equally in qualitative terms, reinforcing the argument that MSE is not always an accurate metric for image quality.

A self-similar model for fretting wear contact with the third body in gross slip

A continuum mechanics approach is applied to model fretting wear in gross-slip regime with accounting for the third-body layer, which is formed by wear debris trapped within contact interface. The developed modeling framework employs the known concept of a third-body conversion factor and utilizes it in a self-similar form. A two-dimensional Hertzian type contact geometry (cylinder on a plate of relatively large thickness) is assumed, though a major part of the presented analysis is curried out for an arbitrary initial gap function. A Winkler foundation model is used to describe the contact deformation of the third-body layer. The effect of elasticity is neglected in describing the self-similar evolution of the contact pressure profile in the advanced stage of fretting wear. A comparison with the FEM-based simulations available in the literature is given. The fact that the elasticity effect plays a minor role has been verified a posteriori in the considered example. Simple formulas are derived to estimate the continuous variation of the third body layer profile, thickness, and volume.

On a possible cosmological evolution of galaxy cluster YX−YSZE scaling relation

An important result from self-similar models that describe the process of galaxy cluster formation is the simple scaling relation YSZEDA2/CXSZEYX= C. In this ratio, YSZE is the integrated Sunyaev-Zel'dovich effect flux of a cluster, its x-ray counterpart is YX, CXSZE and C are constants and DA is the angular diameter distance to the cluster. In this paper, we consider the cosmic distance duality relation validity jointly with type Ia supernovae observations plus 61 YSZEDA2/CXSZEYX measurements as reported by the Planck Collaboration to explore if this relation is constant in the redshift range considered (z<0.5). No one specific cosmological model is used. As basic result, although the data sets are compatible with no redshift evolution within 2σ c.l., a Bayesian analysis indicates that other C(z) functions analyzed in this work cannot be discarded. It is worth to stress that the observational determination of an universal C(z) function turns the YSZEDA2/CXSZEYX ratio in an useful cosmological tool to determine cosmological parameters.

A study of radial self-similar non-relativistic MHD outflow models: parameter space exploration and application to the water fountain W43A

ABSTRACT Outflows, spanning a wide range of dynamical properties and spatial extensions, have now been associated with a variety of accreting astrophysical objects, from supermassive black holes at the core of active galaxies to young stellar objects. The role of such outflows is key to the evolution of the system that generates them, for they extract a fraction of the orbiting material and angular momentum from the region close to the central object and release them in the surroundings. The details of the launching mechanism and their impact on the environment are fundamental to understand the evolution of individual sources and the similarities between different types of outflow-launching systems. We solve semi-analytically the non-relativistic, ideal, magnetohydrodynamics equations describing outflows launched from a rotating disc threaded with magnetic fields using our new numerical scheme. We present here a parameter study of a large sample of new solutions. We study the different combinations of forces that lead to a successfully launched jet and discuss their global properties. We show how these solutions can be applied to the outflow of the water fountain W43A for which we have observational constraints on magnetic field, density and velocity of the flow at the location of two symmetrical water maser emitting regions.

The combined effect of size, inertia and porosity on the indentation response of ductile materials

Herein, we present a self-similar cavity expansion model that follows from the work of Cohen and Durban (2013b) to analyze the dynamic indentation response of elasto-plastic porous materials while accounting for the plastic strain gradient induced size effect. The incorporation of the plastic strain gradient induced size effect in the dynamic cavity expansion model for elasto-plastic porous materials is the key novelty of our model. The material hardness predicted using the cavity expansion model for a wide range of indentation depths and speeds is compared against the available experimental results for OFHC copper, for strain rates varying from 10−4s−1 to 108s−1. We note that despite several simplifying assumptions, the predictions of our cavity expansion model show a reasonable agreement with the experimentally measured material hardness over a wide range of indentation depths and speeds. In addition, we have also carried out parametric analysis to elucidate the specific roles of indentation speed, size effect and initial porosity, on the material hardness and cavitation fields that develop during the indentation process. In particular, our parametric analysis shows that there exists a critical value of the indentation speed beyond which the contribution of inertial effect becomes extremely important and the material hardness increases rapidly. While the influence of the initial porosity on the material hardness is found to increase with increasing indentation speed and decrease with increasing size effect.