Density Evolution Research Articles

Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions

The Fokker–Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker–Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.

A steady-state irradiation creep and thermal creep model for zirconium alloys

In this work, a mechanistic steady-state creep model is developed to characterize the macroscopic strain rate of metallic materials affected by irradiation flux, testing temperature and applied stress. Thereinto, the steady-state strain rate is obtained by considering the density evolution of mobile dislocations, which involve the mechanisms of dislocation multiplication, dynamic annihilation and time-dependent dislocation annihilation. In order to effectively analyze the irradiation creep behavior in the low and high stress region, main attentions are focused on the influence of dislocation mobility on the process of time-dependent dislocation annihilation. At low stress, dislocation mobility is affected by dislocation climb and thermally activated glide. For the former, the absorption of irradiation-induced point defects can promote dislocation climb; whereas, thermally activated dislocation glide might be inhibited by the existing defect clusters. At high stress, the dominant deformation mechanism changes from dislocation climb to displacement cascade unpinning. For the latter, an explicit formula for dislocation mobility is deduced to address the influence of dislocation unpinning from the barriers on irradiation creep. To further verify the established model, thermal creep and irradiation creep data of zirconium alloys are considered to compare with the theoretical results. A good agreement is achieved over a wide range of temperature and stress indicating that the model can well describe the deformation behavior of steady-state creep. In addition, contribution of the dominant dislocation mobility components and dislocation density evolution components is compared under both thermal and irradiation creep, which can facilitate the comprehension of fundamental creep mechanisms of metallic materials.

Investigations of Flaring Plasma Parameters during an M-class Flare Using the Differential Evolution Method and XSM/Chandrayaan-2 Observations

We employ the differential evolution (DE) method to analyze observations from the Solar X-Ray Monitor on board the Chandrayaan-2 spacecraft. DE belongs to the family of evolutionary algorithms that find solutions using mechanisms inspired by biological processes. This approach enables us to simultaneously calculate the distribution of the differential emission measure and elemental abundances through an iterative process. We establish a model for the emission sources of flaring plasma, incorporating temperature, emission measure, and abundances of eight elements: Mg, Al, Si, S, Ar, Ca, Fe, and Ni, for an M3.9 GOES-class solar flare that occurred on 2021 May 7 (SOL2021-05-07T19:04). Our analysis covers various phases of the flare, determining the evolution of temperature, emission measure, and elemental abundances. Additionally, utilizing data from the Spectrometer Telescope for Imaging X-rays on board the Solar Orbiter, we investigate the evolution of hard X-ray source morphology, source volume, electron density, and thermal behavior of the flaring plasma throughout the event. The results reveal notable variations in elemental abundances between photospheric and coronal values during different flare phases. This emphasizes the significance of elemental abundance information in comprehending X-ray emissions during solar flares.

Dynamic evolutions of local current density and water–gas distribution of proton exchange membrane fuel cell with dead-ended anode

Dynamic evolutions of local current density and water–gas distribution of proton exchange membrane fuel cell with dead-ended anode

Current density in a nonlinear medium of carbon nanotubes under the action of diffraction free laser pulses

Based on the spatial distribution of the electric field strength of diffraction-free extremely short Bessel and Airy–Bessel optical pulses, patterns of the temporal evolution of the current density in a nonlinear array of semiconductor carbon nanotubes with a spatial modulation of the refractive index are constructed. The dependences of the maximum value of the current density on the modulation parameters of the refractive index of the medium are established.

An improved cellular automaton model of dynamic recrystallization and the constitutive model coupled with dynamic recrystallization kinetics for microalloyed high strength steels

This study analyzed the deformation behavior of Nb–Ti microalloyed high strength steel at strain rates of 0.001–0.1 s−1 and deformation temperatures of 1050–1150 °C using hot compression tests. The deformation activation energy of 321.1 kJ/mol was calculated based on the stress-strain curve, and the characteristic points on the flow stress curve were extracted. In contrast, the relationship equation between the characteristic points and the Zener-Hollomon parameter was established. A modified dynamic recrystallization (DRX) kinetics model was proposed based on the stress-strain data, considering the strain rate and deformation temperature. A high-precision constitutive model was established based on the dislocation density evolution phenomenological model, which couples with the modified DRX kinetics model. Furthermore, an improved DRX cellular automaton (CA) model was constructed, which comprehensively took into account a series of processes such as dislocation density evolution, recrystallization nucleation, and grain growth. Notably, the model also embedded the pinning effect of second phase (SP) particles on the DRX process. The improved CA model can control the pinning force by adjusting the size and volume fraction of SP particles and can accurately predict the evolution characteristics of DRX and the average austenite grain size of Nb–Ti microalloyed high strength steels during thermal deformation. The development of an improved CA model can be extended to apply to the DRX evolution of other microalloyed steels, and the model provides a powerful tool and method for understanding the thermal deformation behavior and microstructural evolution of microalloyed steels.

Assessment of the Effect of Plasma Density Evolution of the T-15MD Tokamak on the Phase and Frequency Shift of the Signals of the Vertical Channel of the Microwave Interferometer

Assessment of the Effect of Plasma Density Evolution of the T-15MD Tokamak on the Phase and Frequency Shift of the Signals of the Vertical Channel of the Microwave Interferometer

Study of the surface damage threshold and mechanism of KDP crystal under ultrashort laser irradiation

Here, we investigate the mechanism of surface damage threshold anisotropy induced by ultrashort laser in potassium dihydrogen phosphate (KDP) crystal. Carrier-lattice nonequilibrium interaction is simulated based on Brillouin light-scattering (BLS) spectroscopy and a complete self-consistent model to obtain the time evolution of carrier density and temperature as well as lattice temperature. The results indicate that the trend of the lattice temperature is consistent with the experimental phenomena. Meanwhile, the electron-phonon coupling effect, in addition to the electron density traditionally considered, is an important factor affecting damage and is a major contributor to the anisotropy of the damage threshold.

An adaptive hybrid deep learning-based reliability assessment framework for damping track system considering multi-random variables

Reliability assessment is essential in the design and management of rail transit system (RTS), with a heightened focus on damping tracks that feature low-stiffness structures and various uncertainties. However, existing research on RTS has notable gaps: one is the unexplored nonlinear track irregularities in surrogate model, while the other is the significant computational burden of current reliability analysis. Therefore, Firstly, based on a database acquired from simulations of rigid-flexible coupling dynamic models, a hybrid deep learning (DL) surrogate model is developed and trained. Notably, it accommodates both time-series and feature parameters data as inputs, and integrates a novel algorithm featuring slide window with variational decomposition mode-sample entropy, optimization algorithm and adaptive learning strategy (AS). Subsequently, an advanced reliability assessment framework is proposed, utilizing AS hybrid DL-based Probability Density Evolution Method (PDEM), which thoughtfully designed to analyze the two aforementioned categories of random variables and then applied to a train-steel spring floating slab track. Experimental studies show the proposed surrogate model effectively and robustly predicts four safety metrics, with each component's excellence confirmed. Furthermore, this framework shows higher accuracy than the Monte Carlo Method and enhances computational efficiency by 10 ∼ 30 times compared to traditional mechanism-based PDEM. Consequently, reliability assessments indicate values of 0.9312 and 0.9214 for wheel-load reduction rate and rail displacement, respectively, with other metrics showing zero safety hazards. Findings of this study have practical applicability in the field of RTS design and maintenance.

Effective Potential for Quintessential Inflation Driven by Extrinsic Gravity

We numerically study the evolution of the extrinsic energy density in the context of an inflationary regime at the background level in a five-dimensional model using a Bayesian analysis from a dynamic nested sampler (DYNESTY) code. By means of the Nash–Greene embedding theorem, we show that the corresponding model provides an effective potential driven by the influence of extrinsic geometry. We obtain a quintessential inflation that defines a model with a potential V(ϕ)=e−α1ϕ(1−α2ϕ2), where α1 and α2 are dimensionless parameters. Using some known phenomenological parameterizations, such as Chevallier–Polarski–Linder (CPL) and Barboza–Alcaniz (BA) parameterizations, we show that the model reflects a slow-varying inflation preferring a thawing behavior, suggesting an optimistic scenario for further research on the unification of inflation with late cosmic acceleration.

Mean zero artificial diffusion for stable finite element approximation of convection in cellular aggregate formation

We develop and implement finite element approximations for the coupled problem of cellular aggregate formation. The equation governing evolution of cell density is convective in nature, necessitating a modification of standard approaches to circumvent the instabilities associated with standard finite element approximations. To this end, a novel mean zero artificial diffusion approach is proposed, in which the classical artificial diffusion term is replaced by one that is orthogonal to its projection on to continuous functions. The resulting approach for the convective equation is shown to be well-posed. A range of numerical results illustrate the stability and accuracy of the new approach, and its behaviour in comparison with an alternative approach using Taylor–Hood elements. The results also provide insights into the behaviour of cellular aggregates in the context of the model studied here.

A gradient-enhanced bone remodelling approach to avoid the checkerboard phenomenon

Numerical simulation of bone remodelling enables the investigation of short- and long-term stability of bone implants and thus can be an essential tool for surgical planning. The first development of related mathematical models dates back to the early 90’s, and these models have been continuously refined since then. However, one issue which has been under discussion since those early days concerns a numerical instability known as checkerboarding. A literature review of recent approaches guided us to adopt a technique established in damage mechanics and topology optimisation, where similar mesh dependencies and instabilities occur. In our investigations, the so-called gradient enhancement is used to regularise the internal variable field, representing the evolution of the bone mass density. For this, a well-established mathematical model for load-adaptive bone remodelling is employed. A description of the constitutive model, the gradient enhancement extension and the implementation into an open-access Abaqus user element subroutine is provided. Parametric studies on the robustness of the approach are demonstrated using two benchmark examples. Finally, the presented approach is used to simulate a detailed femur model.

Reliability-based design optimization for seismic structures considering randomness associated with ground motions

When addressing the dynamic reliability analysis of structures, it becomes necessary to account for multiple limit state functions or their combinations. In scenarios where structures are subjected to random excitation, this can lead to intricate inter-dependencies among different limit states, and the computational workload can pose a substantial challenge in ensuring sufficient precision. Code-based design primarily ensures safety at the member level, while deterministic optimization fails to accommodate the inherent uncertainties associated with external excitation or the system as a whole. Therefore, in such cases, to address both the uncertainties in excitations and the presence of multiple limit states while mitigating computational challenges, equivalent extreme-value criteria are employed within the framework of the probability density evolution method to calculate the global reliability of the structure subjected to stochastic ground motions generated from the physically motivated stochastic ground motion model. Numerical optimization is subsequently conducted using genetic algorithms, aiming to minimize the cost of the superstructure while adhering to the design performance criteria related to the inter-story drift ratio and considering global reliability. Additionally, multi-objective optimization is carried out using NSGA-II, permitting the generation of multiple solutions, from which one can select the most suitable solution as needed. The numerical results illustrate the effectiveness of this technique in achieving an optimal balance between the cost of the structure and the consideration of global reliability, providing a comprehensive solution for dynamic reliability analysis and design optimization of structures under random excitations.

The effect of electric current on dislocation activity in pure aluminum: A 3D discrete dislocation dynamics study

Introducing electric current into metals and alloys to improve their ductility and to reduce the hardening tendency has been widely adopted. Nevertheless, how the electricity affects the plastic deformation of those materials remains a critical issue with controversial opinions. To clarify the material plastic deformation subjected to electric current, or the so-called electroplastic behavior, an in-depth understanding of the dislocation evolution during that process is vital. From that motivation, three-dimensional dislocation dynamics simulations were carried out to explore the dislocation behavior during the uniaxial tensile deformation of pure aluminum with the introduction of electric current. The scattering process between electrons and dislocations was first captured by a physical model. The electron wind force and local heat effect were quantitatively analyzed by figuring out the momentum and the energy transferred during the interaction of electrons and dislocations respectively. The dislocation density evolution, the activation of different slip systems and the dislocation distribution were further analyzed based on the simulations. The dislocation density in the [111] direction is revealed to increase more significantly than [101] and [001] directions with the introduction of electric current. The results show that the current can reduce the flow stress by promoting the activation of the difficult-to-move dislocations. The activation effect reduces the dislocation tangling tendency and leads to more uniform dislocation distribution. Therefore, a reduction of the flow stress can be observed in EAT comparing to TAT, via the discrete dislocation dynamics simulations even though the dislocation densities are similar. The simulation results are also confirmed by the EAT and TAT experiment results and TEM observations.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

In the recent decade, the reliability analysis of a stochastic system coupled with the uncertainty related to the system’s parameter has attracted much attention. Probability density evolution method (PDEM) is one of the viable options that estimates the probability density function of the structural response by solving generalized density evolution equations (GDEEs). The advantage of PDEM is that it is derived based on the principle of probability conservation, where GDEEs are decoupled from the physical system. In general, GDEEs in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, leading to computationally inefficient for high-fidelity models. With this in view, this study proposes a physics-informed neural network (PINN), a novel deep learning method, based PDEM, for solving the GDEEs. PINN utilizes physical information in the form of differential equations to enhance the performance of the neural networks. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Three numerical examples are presented in this study to illustrate the proposed PINN-based PDEM, including a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resisting frame coupled with a nonlinear energy sink with negative stiffness and sliding friction, and a high-rise timber building coupled with shape memory alloy-based outriggers. The first example is utilized to show the accuracy of the proposed method by comparing results with the Fokker–Planck–Kolmogorov equation and Monte Carlo simulation. The rest two examples are investigated for estimating time-dependent probability of failure. Numerical results show that the proposed PINN-based PDEM can estimate the probability of failure efficiently.

Stochastic stress response and dynamic reliability evaluation for transmission towers with semi-rigid behaviors

As a kind of typical wind-sensitive structure, transmission towers have attracted fast-growing attention in the field of their wind-induced dynamic response. Nevertheless, their dynamic response considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors has not been investigated. To this end, based on the experimental and numerical study, this paper proposes a fitting formula for the stability coefficient of steel tube members with semi-rigid behaviors in transmission towers to determine the dynamic stress response. Then, the stiffness, mass, and damping matrices of steel-tube transmission towers (STTTs) with semi-rigid behaviors are determined to construct their stochastic dynamic finite element model. Subsequently, the integral form of the generalized density evolution equation is solved via a family of Dirac's sequences to conduct the stochastic stress response analysis for STTTs considering effects of semi-rigid connected joints and semi-rigid-constrained stability behaviors, and their dynamic reliability is evaluated by further introducing the extreme-value distribution method. Finally, an engineering example of an existing STTT is given, and the results indicate that the semi-rigid connected joints and semi-rigid-constrained stability behaviors would significantly affect the stochastic stress response and dynamic reliability of STTTs. Accordingly, taking into account semi-rigid connected joints and semi-rigid-constrained stability behaviors may be more applicable for analysis and design of STTTs.

Overall seismic reliability analysis of aqueduct structure based on different levels under random earthquake

The seismic reliability analysis method of large aqueduct structure is of great significance to its operation and maintenance safety evaluation and seismic toughness evaluation in the life cycle. In this paper, the seismic reliability analysis method of aqueduct structure at different levels is proposed based on the probability density evolution theory and the random ground motion model of engineering physics. Taking Shuangji River aqueduct as an example, 100 random seismic acceleration time-history samples and corresponding probability assignments are generated based on the engineering physical random ground motion model, and combined with the nonlinear response analysis platform of fiber beam element of aqueduct structure，the dynamic response analysis of the aqueduct structure under nonlinear and random coupling conditions is carried out from the perspective of the whole process of nonlinear development. And by introducing different failure criteria of the aqueduct structure, the overall seismic resistance of the aqueduct structure under different fortification targets is quantitatively analyzed. The results show that the coupling of randomness and nonlinearity has a significant impact on the dynamic response of the aqueduct structure. The overall seismic reliability analysis method based on different fortification levels can accurately solve and evaluate the overall seismic reliability of the aqueduct structure under different fortification levels. The research results have important theoretical significance for the overall seismic reliability analysis and design of aqueduct structures based on multi-level fortification targets.

Dispersive Modeling of Normal and Cancerous Cervical Cell Responses to Nanosecond Electric Fields in Reversible Electroporation Using a Drift-Step Rectifier Diode Generator.

This paper creates an approximate three-dimensional model for normal and cancerous cervical cells using image processing and computer-aided design (CAD) tools. The model is then exposed to low-frequency electric pulses to verify the work with experimental data. The transmembrane potential, pore density, and pore radius evolution are analyzed. This work adds a study of the electrodeformation of cells under an electric field to investigate cytoskeleton integrity. The Maxwell stress tensor is calculated for the dispersive bi-lipid layer plasma membrane. The solid displacement is calculated under electric stress to observe cytoskeleton integrity. After verifying the results with previous experiments, the cells are exposed to a nanosecond pulsed electric field. The nanosecond pulse is applied using a drift-step rectifier diode (DSRD)-based generator circuit. The cells' transmembrane voltage (TMV), pore density, pore radius evolution, displacement of the membrane under electric stress, and strain energy are calculated. A thermal analysis of the cells under a nanosecond pulse is also carried out to prove that it constitutes a non-thermal process. The results showed differences in normal and cancerous cell responses to electric pulses due to changes in morphology and differences in the cells' electrical and mechanical properties. This work is a model-driven microdosimetry method that could be used for diagnostic and therapeutic purposes.

Stochastic analysis and reliability assessment of critical RC structural components considering material properties uncertainty

The unavoidable heterogeneity in the mechanical characteristics of concrete is widely acknowledged. Although it is widely considered as either perfectly correlated or entirely independent random variables in engineering practice; however, such treatment is illogical, and the outcomes may be deceptive. In high-rise buildings comprised of multiple structural components, it is crucial to consider the material properties’ spatial variability (MPSV) to obtain a reliable structural response and avoid damage to these structures. To this end, three main components, including column, rectangular shear wall, and U-shaped shear wall, are considered herein to investigate their stochastic response. The MPSV is represented by a covariance matrix decomposition-based random field generator combined with a GF-discrepancy-based point selection strategy to generate samples efficiently. A simplified strategy is developed to represent the random field for the U-shaped wall. Moreover, the probability density evolution method combined with the extreme value event is employed to obtain the failure probability of the studied components, where failure probabilities of 18%, 23%, and 32% are recorded for the studied RC column, rectangular shear wall, and U-shaped shear wall, respectively. Furthermore, different failure modes were identified and could not be determined through the deterministic analysis, highlighting the importance of accounting for material uncertainty. The proposed framework proved that the stochastic response and non-linear behavior of the considered components could be well captured and provide full perspective about the uncertainty quantification and reliability assessment and can be further implemented to capture the stochastic response and safety assessment of high-rise buildings.

Gas density evolution and recirculation after a nanosecond pulsed discharge: experimental and computational results

Gas density evolution and recirculation after a nanosecond pulsed discharge: experimental and computational results