Discrete Effects Research Articles

Analyzing Pillar Strength and Behavior using Wolfram Mathematica code: Effects of Cracks, Size and Discretization

The stability of fractured rock masses and the modeling of underground mining pillars necessitate a comprehensive understanding of behavioral models and mechanical properties. This study employs the Wolfram Mathematica code to investigate mining pillar reliability, specifically focusing on elucidating the influence of scale and shape on pillar strength. Drawing inspiration from methodologies in the existing literature, our approach is based on the Mohr-Coulomb theory and Griffiths's random field of rock strength. This study highlights the significance of shape, where pillar strength exhibits exponential growth with increasing width-to-height ratios. Beyond a critical value, strength surges, especially under elevated confining stress. Additionally, a critical mesh size significantly affects the weakest pillar behavior. Our results confirm the 'size effect,' wherein strength generally decreases with increasing pillar volume. Thus, strength decreases with rising volume until a threshold. Particularly noteworthy is the phenomenon observed in the presence of cracks; initially, an increase in mesh size leads to a decline in strength, corresponding to an increase in the number of cracks. However, this decline stabilizes beyond a critical mesh size, after which strength experiences a resurgence echoing behavior seen in the homogeneous case. In this paper, the reproduction of the scale effect by an algorithm based on the Mathematica code was made to allow a probabilistic study to be carried out because of the random existence of discontinuities in nature – another hand to carry out stochastic modeling of fractures and its influence on the rock mass strength.

A hybrid tau-leap for simulating chemical kinetics with applications to parameter estimation.

We consider the problem of efficiently simulating stochastic models of chemical kinetics. The Gillespie stochastic simulation algorithm (SSA) is often used to simulate these models; however, in many scenarios of interest, the computational cost quickly becomes prohibitive. This is further exacerbated in the Bayesian inference context when estimating parameters of chemical models, as the intractability of the likelihood requires multiple simulations of the underlying system. To deal with issues of computational complexity in this paper, we propose a novel hybrid τ-leap algorithm for simulating well-mixed chemical systems. In particular, the algorithm uses τ-leap when appropriate (high population densities), and SSA when necessary (low population densities, when discrete effects become non-negligible). In the intermediate regime, a combination of the two methods, which uses the properties of the underlying Poisson formulation, is employed. As illustrated through a number of numerical experiments, the hybrid τ offers significant computational savings when compared with SSA without, however, sacrificing the overall accuracy. This feature is particularly welcomed in the Bayesian inference context, as it allows for parameter estimation of stochastic chemical kinetics at reduced computational cost.

Discreteness effects in the postreconstruction galaxy power spectrum

Discreteness effects in the postreconstruction galaxy power spectrum

Travelling waves for discrete stochastic bistable equations

Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise. In this paper we study the combined effect of spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation, which is a prototypical model for bistable reaction-diffusion partial differential equations (PDEs). We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probability if the noise covariance and spatial discretization are sufficiently small.

QCD in the chiral SU(3) limit from baryon masses on lattice QCD ensembles

The baryon masses on Coordinate Lattice Simulations (CLS) ensembles are used to determine the Low-Energy Constants (LEC) that characterize quantum chromodynamics in the flavor-SU(3) limit with vanishing up, down, and strange quark masses. Here we reevaluate some of the baryon masses on flavor-symmetric ensembles with much-improved statistical precision, in particular for the decuplet states. These additional results then lead to a more significant chiral extrapolation of the lattice dataset to its chiral SU(3) limit. Our results are based on the chiral Lagrangian with baryon octet and decuplet fields considered at the one-loop level. Finite-box and discretization effects of the lattice data are considered systematically. While in our global fit of the data we insist on large-Nc sum rules for the LEC that enter at N3LO, all other LEC are unconstrained. In particular, we obtain values for the chiral limit of the pion decay constant and the isospin-limit of the quark-mass ratio compatible with the FLAG report. Published by the American Physical Society 2024

Well-posedness and finite element analysis for the elastic scattering problem with a modified DtN map

As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application, and then the well-posedness of the resulting boundary value problem (BVP) becomes a challenging issue. By introducing a corresponding eigensystem to the bilinear form together with appropriate truncated norm estimates, we prove the well-posedness of the corresponding BVP in a weak sense. In addition, a priori error estimates that incorporate the effects of the finite element discretization and the truncation of infinite series are derived. Finally, numerical tests are implemented to validate the theoretical results.

A strategy for scaling the hardening behavior in finite element modelling of geometrically exact beams

AbstractA yield function in the stress resultant space of geometrically exact beams based on the elastoplastic cross-sectional warping problem has been proposed by Herrnböck et al. (Comput Mech, 67(3):723–742, 2021). This plasticity framework has been extended with a hardening tensor to model the kinematic hardening effects in Herrnböck et al. (Comput Mech, 71(1):1–24, 2022). While this framework provides scaling for the yield surface in ideal plasticity, scaling in hardening plasticity has not yet been explored. This paper focuses on the numeric modelling of hardening beams and beam assemblies at different geometric scales. Discretization effects from the introduction of plasticity into the geometrically exact beam model are demonstrated. Furthermore, the effects of scaling are explored, and a method to mitigate undesirable effects in order to achieve a size-agnostic formulation is proposed. Consistent geometric scaling is demonstrated for two alternative scaling approaches of the yield function.

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

It's about (taking up) space: Discreteness of individuals and the strength of spatial coexistence mechanisms.

One strand of modern coexistence theory (MCT) partitions invader growth rates (IGR) to quantify how different mechanisms contribute to species coexistence, highlighting fluctuation-dependent mechanisms. A general conclusion from the classical analytic MCT theory is that coexistence mechanisms relying on temporal variation (such as the temporal storage effect) are generally less effective at promoting coexistence than mechanisms relying on spatial or spatiotemporal variation (primarily growth-density covariance). However, the analytic theory assumes continuous population density, and IGRs are calculated for infinitesimally rare invaders that have infinite time to find their preferred habitat and regrow, without ever experiencing intraspecific competition. Here we ask if the disparity between spatial and temporal mechanisms persists when individuals are, instead, discrete and occupy finite amounts of space. We present a simulation-based approach to quantifying IGRs in this situation, building on our previous approach for spatially non-varying habitats. As expected, we found that spatial mechanisms are weakened; unexpectedly, the contribution to IGR from growth-density covariance could even become negative, opposing coexistence. We also found shifts in which demographic parameters had the largest effect on the strength of spatial coexistence mechanisms. Our substantive conclusions are statements about one model, across parameter ranges that we subjectively considered realistic. Using the methods developed here, effects of individual discreteness should be explored theoretically across a broader range of conditions, and in models parameterized from empirical data on real communities.

Effect of discretization on off-axis listeners in binaural synthesis based on the optimal source distribution

3D sound reproduction based on binaural synthesis over loudspeakers is one of the most important techniques in immersive spatial audio. The system inversion involved in the binaural synthesis gives rise to several problems such as dynamic range loss, distortion of the reproduction transfer function, and a lack of robustness to room reflections. The principle of Optimal Source Distribution was proposed to overcome these problems. This utilizes a pair of conceptual monopole transducers whose azimuthal location varies continuously as a function of frequency. Another potential of the principle is that it can provide listening positions for multiple off-axis listeners, in addition to the on-axis listener whose inverse matrix is given, since the phase interference state is aligned at all frequencies. However, these continuously varying transducers are too conceptual to realize in practice. It is necessary to provide discretized sound sources with appropriate frequency ranges according to the principle. In this paper, the effect of discretization of the principle on off-axis listeners' performance is numerically investigated to reveal how those off-axis sweet spots are formed as the discretization becomes finer. An experiment is also carried out using an actual discretized system and the effect of the head-related transfer functions is considered.

Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network

Fixed point lattice actions are designed to have continuum classical properties unaffected by discretization effects and reduced lattice artifacts at the quantum level. They provide a possible way to extract continuum physics with coarser lattices, thereby allowing one to circumvent problems with critical slowing down and topological freezing toward the continuum limit. A crucial ingredient for practical applications is to find an accurate and compact parametrization of a fixed point action, since many of its properties are only implicitly defined. Here we use machine learning methods to revisit the question of how to parametrize fixed point actions. In particular, we obtain a fixed point action for four-dimensional SU(3) gauge theory using convolutional neural networks with exact gauge invariance. The large operator space allows us to find superior parametrizations compared to previous studies, a necessary first step for future Monte Carlo simulations and scaling studies. Published by the American Physical Society 2024

Spectrum of global string networks and the axion dark matter mass

Cold dark matter axions produced in the post-inflationary Peccei-Quinn symmetry breaking scenario serve as clear targets for their experimental detection, since it is in principle possible to give a sharp prediction for their mass once we understand precisely how they are produced from the decay of global cosmic strings in the early Universe. In this paper, we perform a dedicated analysis of the spectrum of axions radiated from strings based on large scale numerical simulations of the cosmological evolution of the Peccei-Quinn field on a static lattice. Making full use of the massively parallel code and computing resources, we executed the simulations with up to 112643 lattice sites, which allows us to improve our understanding of the dependence on the parameter controlling the string tension and thus give a more accurate extrapolation of the numerical results. We found that there are several systematic effects that have been overlooked in previous works, such as the dependence on the initial conditions, contaminations due to oscillations in the spectrum, and discretisation effects, some of which could explain the discrepancy in the literature. We confirmed the trend that the spectral index of the axion emission spectrum increases with the string tension, but did not find a clear evidence of whether it continues to increase or saturates to a constant at larger values of the string tension due to the severe discretisation effects. Taking this uncertainty into account and performing the extrapolation with a simple power law assumption on the spectrum, we find that the dark matter mass is predicted in the range of m a ≈ 95–450 μeV.

Discrete time stability of augmented Lagrangian formalism based underactuated inverse dynamics control method

Stability problems of robotic systems arise sometimes suddenly, seemingly for no reason. The digital time sampling is often the main cause of these instabilities. Discrete models, which are capable of the prediction of stability, are available for low-degree-of-freedom template models of linear position and force control. However, for the inverse dynamics control of underactuated systems, the literature has a lack of generally applicable results related to the effect of time discretization. Several control approaches are available in the literature out of which a widely used one, the augmented Lagrangian formalism and its stability properties are analysed in this work. Theoretical stability properties are obtained for a generally usable, linear, underactuated, two-degree-of-freedom constrained template model. The actuator dynamics, the finite difference approximation of the feedback velocity and the filtering of the feedback data are considered in the model. These phenomena strongly affects the stability properties. The theoretically obtained stability maps are experimentally validated on an underactuated crane-like indoor robot. The position and orientation accuracy of the robot were assessed: the absolute position error was below 30 mm and the orientation error was below 3°.

Absorbing discretization effects with a massive renormalization scheme: The charm-quark mass

We present the first numerical implementation of the massive symmetric-momentum-subtraction (mSMOM) renormalization scheme and use it to calculate the charm-quark mass. Based on ensembles with three flavors of dynamical domain-wall fermions with lattice spacings in the range 0.11–0.08 fm, we demonstrate that the mass scale which defines the mSMOM scheme can be chosen such that the extrapolation has significantly smaller discretization effects than the SMOM scheme. Converting our results to the MS¯ scheme we obtain m¯c(3 GeV)=1.008(13) GeV and m¯c(m¯c)=1.292(12) GeV. Published by the American Physical Society 2024

Global spectrum model of discrete dislocation equation

The fully discrete dislocation equation can be rationally derived from the spectrum model. In this paper, the previous spectrum model is reinterpreted to cover the global feature of the spectrum. Instead of the local behavior around the origin of the Brilouin zone, the global spectrum model focuses the maximum at the boundary of the Brillouin as well as the slope at the origin of the Brilouin zone. The strength of the point-contact interaction that effectively describes the discreteness effect becomes larger because the whole lattice effect is included in the global spectrum model. The validness and signification of the new model are illustrated by a solvable lattice dynamical model of the cubic lattice.

Evaluating the effects of nonlocality and numerical discretization in peridynamic solutions for quasi-static elasticity and fracture

The difference between the computational peridynamic results and the corresponding exact classical solutions is contributed by the error induced by numerical discretization and the nonlocality-induced difference. To evaluate and compare these contributions and investigate their dependence on the peridynamic influence functions, in this paper, we apply different peridynamic influence functions and different horizon factors (m, the ratio between the horizon size and the grid spacing) to conduct static (or quasi-static) tensile numerical tests. We calculate the difference between the peridynamic solutions and the corresponding classical solutions for static uniaxial tension of thin plates with or without hole, the J-integral of a single crack under Mode I loading condition, and the quasi-static fracture in a perforated thin plate. For the case of uniaxial tension in a thin, homogeneous plate, we separate the effects of nonlocality and numerical discretization by implementing the boundary conditions in different ways. The numerical results show that both the effects of nonlocality and numerical discretization correspond to the nonlocal constants of the influence functions (with few exceptions). For problems with the presence of the peridynamic surface effect, such as holes, influence function with weaker nonlocality is a better choice to obtain more accurate results.

Charge regulation of nanoparticles in the presence of multivalent electrolytes.

We explore the charge regulation (CR) of spherical nanoparticles immersed in an asymmetric electrolyte of a specified pH. Using a recently developed reactive canonical Monte Carlo (MC) simulation method, titration isotherms are obtained for suspensions containing monovalent, divalent, and trivalent coions. A theory based on the modified Poisson-Boltzmann approximation, which incorporates the electrostatic ion solvation free energy and discrete surface charge effects, is used to compare with the simulation results. A remarkably good agreement is found without any fitting parameters, both for the ion distributions and titration curves, suggesting that ionic correlations between coions and hydronium ions at the nanoparticle surface play only a minor role in determining the association equilibrium between hydroniums and the functional sites on the nanoparticle surface. On the other hand, if suspension contains multivalent counterions, we observe a large deviation between theory and simulations, showing that the electrostatic correlations between counterions and hydronium ions at the nanoparticle surface are very significant and must be properly taken into account to correctly describe CR for such solutions.

A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness

This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.

A novel 4-DOF marine stern bearing support model considering discrete distribution effects

To address the issue of the traditional support model's inability to accurately describe the non-uniform dynamics behavior inside the bearing, this study proposes a novel 4-degree-of-freedom (DOF) bearing support model that incorporates the discrete distribution effects induced by the bi-directional misalignment and bending deformation of the journal. Additionally, a coupling algorithm is designed to study the interaction behaviors between the marine stern bearing lubrication performance and the shaft dynamics. Moreover, a full-size marine shaft alignment test rig for measuring dynamic bearing loads is designed to verify the validity of the present model. On this basis, the effects of different bearing models, rotational speeds, external loads, and bush elastic modulus on the shaft dynamic alignment and the static and dynamic characteristics of the stern bearing are investigated. The experimental results indicate that the 4-DOF model, which takes into account discrete distribution, exhibits a more precise prediction of the alignment characteristics of the shaft system, with a forecast error for the stern bearing load of 1.32%. Heavy loads or low rotational speeds significantly deteriorate the edge-loading effect and increase the stiffness and damping coefficient of the bearings. The bearing lubrication performance directly affects the alignment performance of the shaft system, particularly in low-speed conditions. A softer bush material can increase the minimum oil film thickness, carrying region, and damping coefficient, mitigate the phenomenon of edge-loading, and facilitate the vibration control of the shaft. Research results have made a great contribution to marine shaft alignment optimization and structure design.

Therapeutic Targets in Innate Immunity to Tackle Alzheimer's Disease.

There is an urgent need for effective disease-modifying therapeutic interventions for Alzheimer's disease (AD)-the most prevalent cause of dementia with a profound socioeconomic burden. Most clinical trials targeting the classical hallmarks of this disease-β-amyloid plaques and neurofibrillary tangles-failed, showed discrete clinical effects, or were accompanied by concerning side effects. There has been an ongoing search for novel therapeutic targets. Neuroinflammation, now widely recognized as a hallmark of all neurodegenerative diseases, has been proven to be a major contributor to AD pathology. Here, we summarize the role of neuroinflammation in the pathogenesis and progression of AD and discuss potential targets such as microglia, TREM2, the complement system, inflammasomes, and cytosolic DNA sensors. We also present an overview of ongoing studies targeting specific innate immune system components, highlighting the progress in this field of drug research while bringing attention to the delicate nature of innate immune modulations in AD.