Diffusion Term Research Articles

Data-driven machine learning approach based on physics-informed neural network for population balance model

Population balance models (PBM) are a fundamental tool in the field of process engineering and materials science for understanding and forecasting particulate system dynamics. These models are essential for the design and optimization of a wide range of industrial processes because they capture the evolution of particle size distribution in response to different phenomena such as nucleation, growth, aggregation, and breakage. The complex and non-linear nature of these models makes it difficult to find the analytical solution and even sometimes the approximate solution. This paper introduces a novel machine learning approach based on physics-informed neural network (PINN) for the approximate solution of PBM. Our strategy utilizes the use of a customized neural network framework that has been developed and trained on data generated from simulated PBM by using a finite difference method for the differential operators to identify the underlying dynamics and patterns controlling the evolution of particle distribution. In general, PINN uses automatic differentiation for the computation of differential operators, which is based on the chain rule and needs several matrix operations for computing, which reduces the processing efficiency during the training. The PINN approach provides a flexible, effective and highly adaptable solution framework by utilizing neural networks to approximate the solution and defining the loss function as the sum of the differential equation’s residuals at specific random or regular points inside the domain, as well as the residuals of the initial and boundary conditions. It is shown numerically that this approach does not need a diffusion term for a stable solution, which is often needed in most numerical methods for solving these models. For further validation, we compare the results with the exact solution and find them with a very good agreement with each other.

Multivalued Random Dynamics of Fractional Reaction–Diffusion–AdvectionEquations Driven by Superlinear Colored Noise on Unbounded Domains

This article investigates the multivalued random dynamics generated by solution operators of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains. The equations under consideration exhibit non-Lipschitz continuous nonlinear drift and diffusion terms, which lead to the non-uniqueness of solutions and hence generate multivalued random dynamical systems. Our goals are to demonstrate: (i) the existence of solutions for fractional reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains; and (ii) the existence and uniqueness of pullback random attractors for these systems. The measurability of the random attractors is demonstrated using a method that relies on the weak upper semicontinuity of the solutions. The asymptotic compactness of solution operators is obtained by employing Ball’s approach to energy equations in order to address the non-compactness of Sobolev embeddings on the entire space.

Advancing Hybrid Numerical Methods for Nonlinear Stochastic Differential Equations: Applications in Complex Systems

The focus of this work is to consider composite numerical techniques for the approximation of SDEs with nonlinear coefficients in the drift and diffusion terms. SDEs, crucial for modeling systems with stochastic components, contain nonlinear terms that cause analytical solvability, numerical stiffness, and sensitivity to noise. These difficulties pose a problem for traditional techniques such as Euler-Maruyama or Milstein schemes, specifically in stiff or very nonlinear systems. Accompanying exact methods are numerical methods that include a deterministic synthesis of drift terms and a stochastic interpolation of diffusion terms with the purpose of increasing precision and stability and optimizing used computing time. Discussed approaches include implicit-explicit (IMEX) schemes, spectral collocation methods, and machine learning-assisted techniques. IMEX methods handle stiffness in nonlinear drift terms implicitly, while explicitly handling stochastic diffusion. Spectral-collocation methods utilize high-order polynomial approximations for accuracy in discretization where solutions are smooth and defined in a bounded domain. The combination of these techniques and machine learning extends SDE analysis and concentrates on SDE nonlinearities as well as adaptive solution strategies. They find use in every area of discipline, such as stochastic volatility models in finance, population dynamics in biology, and turbulent fluid flows in engineering. Simulation results show that hybrid schemes outperform other methods in terms of accuracy, stability, and computational expense. This work outlines how the integration of the suggested methods can overcome the shortcomings of the classic approaches so as to enable progression in solving complex, high-dimensional, and nonlinear stochastic problems. Subsequent studies will continue to investigate additional adaptive frameworks and more domain-specific and machine learning-based improvements to expand the spectrum of hybrid use.

Hyperbolic Sine Function Control-Based Finite-Time Bipartite Synchronization of Fractional-Order Spatiotemporal Networks and Its Application in Image Encryption

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results.

Efficient Method for Improving Fully Implicit Scheme in Smoothed Particle Hydrodynamics for Highly Viscous and Non‐Newtonian Polymer Flow

ABSTRACTSmoothed particle hydrodynamics (SPH), a fully Lagrangian particle method, has been widely used in various applications, such as the simulation of incompressible flow with a free surface. For highly viscous flow, which typically appears in polymer processing, a fully implicit scheme, which implicitly treats both the predictor and corrector procedures of the projection method in the SPH framework, is efficient and numerically stable. This study proposes a simple and efficient fully implicit scheme that improves predictive accuracy for highly viscous flow. The proposed scheme utilizes a recently developed scheme for highly viscous flow that performs the pressure calculation before the viscosity diffusion calculation, unlike the conventional projection method. This study further improves the viscosity diffusion calculation by considering a viscous diffusion term that is usually ignored due to the continuity equation. Including this term enables us to correctly perform calculations for non‐Newtonian flows. The increase in the computational cost that results from the inclusion of the term is alleviated by introducing a two‐step method for calculating the viscous diffusion terms. The developed scheme is applied to injection molding of a polymer melt between two parallel plates. The results show that the proposed two‐step implicit scheme reproduces the fountain flow behavior near the advancing front. In addition, a three‐dimensional molding simulation of a plate with a hole shows a significant improvement in the prediction of the filling pattern in the mold.

Discovering stochastic basin stability from data in a Filippov competition system with threshold control.

The existing research studies on the basin stability of stochastic systems typically focus on smooth systems, or the attraction basins are pre-defined as easily solvable regular basins. In this work, we introduce a new framework to discover the basin stability from state time series in the non-smooth stochastic competition system under threshold control. Specifically, we approximate the drift and diffusion with threshold control parameters by an extended Kramers-Moyal expansion with initial state partitioning. Then, we calculate the first transition probability of irregular attraction basins by applying a difference scheme and smooth approximation methods for the system. Numerical simulations of the original system validate the accuracy of the identified drift and diffusion terms, as well as the smooth approximation. Our findings reveal that threshold control modifies the influence of environmental noise on the system basin stability.

An Improved Single‐Layer Smoothed Particle Hydrodynamics Model for Water–Soil Two‐Phase Flow

ABSTRACTIn coastal and offshore engineering, the intense water–soil motion poses significant challenges to the safety of buildings and structures. The smoothed particle hydrodynamics (SPH) method, as a mesh‐free Lagrangian solver, has considerable advantages in the numerical resolution of such problems. SPH models for the water–soil two‐phase flow can be categorized into the multilayer type and the single‐layer type. Although the single‐layer model envisions a simpler algorithm and higher computational efficiency, its accuracy, stability, and recovery of interfacial details are far from satisfactory. In the present work, an improved single‐layer model is established to alleviate these limitations. First, the soakage function, which takes effect near the phase interface, is introduced to characterize the two‐phase coupling status. Additionally, the stress diffusion term and a modified density diffusion term applicable in density discontinuity scenario are introduced to ease the numerical oscillation. Finally, to remove the unphysical voids in the interfacial region, the particle shifting technique with special treatment tailored for free‐surface particles is implemented. Validations of the proposed model are carried out by a number of numerical tests, including the erodible dam‐break problem, the wall‐jet scouring, the flushing case, and the water jet excavation. Appealing agreements with either experimental data or published numerical results have been achieved, which verifies the accuracy, stability, and robustness of the proposed model for water–soil two‐phase flows.

Free evolution in the Ginzburg-Landau equation and other complex diffusion equations

Abstract New ordinary differential equations (ODEs) for the evolution of spectral components are derived from the complex Ginzburg-Landau equation (CGLe) for one-dimensional spatial domains without boundaries (free evolution) and with one fixed boundary (semi-free evolution). For such evolution, a complex or imaginary diffusion term creates a tendency for waves to lengthen. This requires a novel ansatz and auxiliary condition that treat wavenumbers as time-varying. The ansatz consists of a discrete spatial Fourier transform modified with a time-dependent wavenumber for the peak spectral component. The wavenumbers of the other components are fixed relative to this wavenumber. The new auxiliary condition is the terminal condition for complex diffusion (after wavenumbers evolve to zero, they remain at zero). The derived free and semi-free ODEs are solved along characteristic lines located symmetrically about a fixed spatial point. Waves lengthen with time away from this point in both directions. Laboratory experiments on the formation of channel sandbars, theoretically described by the CGLe, show two regions whose evolutionary behaviour is qualitatively predicted by the free and semi-free evolution equations. This analysis applies to other time-dependent partial differential equations with complex or imaginary diffusion terms. New freely evolving solutions are derived for the complex heat equation and Schrödinger equation (linear and nonlinear).

A finite element approach to the Keller–Segel Chemotaxis system with impact of nonlocal diffusion

This paper presents the investigation of a boundedness solution for a Keller–Segel chemotaxis (KSC) system with nonlocal diffusion. In order to achieve boundedness, we derive the necessary and sufficient condition based on the assumption of ensuring a unique classical global bounded solution for the proposed KSC system. In this work, the proposed system’s boundedness condition is determined based on three particular situations: when χ<χ∗, χ<κ, and when χ=κ. Finally, the effectiveness of the nonlinear and nonlocal diffusion terms in the KSC system is demonstrated through numerical simulations under consideration of the values of system parameters. From the simulation results, we can confirm the effectiveness of the proposed method.

Traveling wave fronts for a nonlocal system with delays in both diffusion and reaction terms

Traveling wave fronts for a nonlocal system with delays in both diffusion and reaction terms

Stability Analysis of Split‐Step θ$$ \theta $$‐Milstein Scheme for Stochastic Delay Integro‐Differential Equations

ABSTRACTThis paper introduces and analyzes a split‐step ‐Milstein (SSTM) scheme specifically designed for stochastic delay integro‐differential equations (SDIDEs). Given reasonable assumptions regarding the drift term, diffusion term, and integral term, it is established that for small stepsize constraint, the SSTM method with is mean square exponential stable. Additionally, the result shows that also demonstrates the same stability under more stringent conditions on stepsize. Ultimately, two numerical experiments are presented to verify the validity of the obtained results.

Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations

It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems.

Meshfree methods for nonlinear equilibrium radiation diffusion equation with interface and discontinuous coefficient

The partial differential equation describing equilibrium radiation diffusion is strongly nonlinear, which has been widely utilized in various fields such as astrophysics and others. The equilibrium radiation diffusion equation usually appears over multiple complicated domains, and the material characteristics vary between each domain. The diffusion coefficient near the interface is discontinuous. In this paper, the equilibrium radiation diffusion equation with discontinuous diffusion coefficient will be solved numerically by the unsymmetric radial basis function collocation method. The energy term T4 is linearized by utilizing the Picard-Newton and Richtmyer linearization methods on the basis of the fully implicit scheme discretization. And the successive permutation iteration and direct linearization methods are applied to linearize the diffusion terms. The accuracy of the proposed methods is proved by numerical experiments for regular and irregular domains with different types of interfaces.

Coupled in-line and cross-flow vortex-induced vibration responses of a fluid-conveying riser with variable tension in shear flow

When seawater streams around risers, it causes vortex-induced vibrations (VIV), which occur in two forms: in-line (IL) and cross-flow (CF). Accurate prediction of coupled IL and CF VIV behaviors is essential for designing risers. To investigate the VIV of marine risers used in deep-sea oil and gas transportation, this study analyzed the coupled CF and IL VIV characteristics of the riser with axially time-varying tension under the combined effects of internal flow and oceanic linear shear flow. The work established the vibration control equation for the riser considering internal flow velocity, axial top tension, and bending stiffness, which is based on Euler-Bernoulli beam theory. The double Van der Pol diffusion wake oscillator model was used to simulate the vortex-induced forces from the ocean currents, and the internal fluid was considered as a single-phase incompressible liquid. Using the Generalized Integral Transform Technique (GITT), the coupled system of nonlinear partial differential equations was further transformed into a system of nonlinear ordinary differential equations for numerical solution. A parametric study was conducted to analyze the impact of current velocity, internal flow velocity, and diffusion term on the VIV responses, including structural displacement, structural frequency, displacement envelope, and displacement evolution. Numerical results indicate that the vibration modes of the riser are influenced by both CF and IL directions, and the effect of IL can not be ignored. The diffusion term has a significant impact on the vibrations of the riser. The number of vibration modes of the riser is mainly influenced by the increasing current velocity, and for a given current velocity, the vibration of the riser becomes chaotic when the dimensionless internal fluid velocity increased within a certain range.

Conjugated Heat Transfer of the stator of an synchrounous electrical machine

Electrical energy generation in remote islands and ships are mainly due the energy conversion from chemical (fuel) to mechanical by an engine and converted from mechanical to electrical by an alternator. The losses in the alternator during the energy conversion are responsible for the increase of its temperature. In order to control the temperature level of the equipment, it is necessary to control how heat is evacuated. The present work numerically evaluates the heat dissipation in the stator of an electrical machine and its temperature maximum levels. Heat is generated in the cooper (windings) and steel (stator main body) during the energy conversion and is evacuated due the forced airflow throughout the stator. In order to reduce computational costs, due geometrical symmetry, one quarter of the machine was modeled. In addition to symmetry, the momentum boundary conditions consist in inlet, outlet and noslip at the walls and the thermal boundary conditions consist in imposed heat fluxes. In order to isolate the inlet and outlet boundary conditions, they were positioned at least 10 times the respective channel height far from the stator. The model is transient and the initial flow field is at rest (0 m/s) and at ambient temperature (300 K). The geometry was constructed using Salome-Meca software and the mesh was generated using the snappyHexMesh solver from OpenFOAM v11. The foamMultiRun solver from OpenFOAM was employed to create the numerical model. It solves the mass, momentum and energy equations in transient regime. The numerical schemes employed were linearUpwind (second order) for the advective terms, Gauss for the diffusive terms (linear) and the backward time discretization (second order). The turbulence model employed was the K-omega SST. The model predicted where were the regions inside the stator with intese heat transfer, which implies in lower temperature magnitudes and also where there were temperature peaks. Then correlation between the temperature and flow fields were discussed.

Calculation of closed combustion performance based on evolution of gun propellant burning surface

Abstract In order to enhance the combustion performance evaluation ability of irregularly shaped gun propellants and accelerate the development of propellant energy-release control technology, the present work provides the calculation method of closed combustion performance based on the evolution of gun propellant burning surface. The 3-D structure of the gun propellant should be built, and then a steady-state heat conduction equation with a source term is set, which can be considered as the Poisson equation with a diffusion term. The variable in the equation is the burned web thickness of the propellant grain. Afterwards, finite element meshing was performed on the 3-D structure to obtain a dynamic database of equivalent surfaces with burned web size. The pressure during the closed combustion process can be obtained from the mass conservation equation as a function of the burned web thickness. Furthermore, the combustion time field can be derived based on the functional relationship between burning-rate and pressure. Ultimately, the variation of gas pressure versus time in the closed bomb could be obtained. The results showed that the burnout time of the tubular propellant was 1.6 ms shorter than that of the inner pore offset 0.5 mm under the design conditions. This research can provide researchers with a new method for calculating the closed combustion performance of irregular gun propellants, which can be applied to the design of propellant shapes and the study of launch safety of propellant charges.

Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

Error analysis of implicit-explicit weak Galerkin finite element method for time-dependent nonlinear convection-diffusion problem

This paper focuses on the exploration of an implicit-explicit (IMEX) weak Galerkin finite element method (WG-FEM) applied to a one-dimensional nonlinear convection-diffusion equation. Based on a special weak form featuring two built-in parameters, we propose the fully implicit-explicit discrete WG finite element scheme. The diffusion term is treated implicitly, while the nonlinear convection term is treated explicitly. The WG-FEM utilizes locally piecewise polynomials of degree k to approximate the primal variable within the element interiors, along with piecewise polynomials of degree k+1 for the weak derivatives. Optimal error estimates in the L2 norm for the fully discrete scheme are derived in the theoretical analysis. Furthermore, we conduct numerical experiments to illustrate the effectiveness and accuracy of the proposed scheme.

Convergence analysis of exponential time differencing scheme for the nonlocal Cahn–Hilliard equation

In this paper, we provide a rigorous proof of the convergence for both first-order and second-order exponential time differencing (ETD) schemes applied to the nonlocal Cahn–Hilliard (NCH) equation. The spatial discretization is executed through the Fourier spectral collocation method, whereas the temporal discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation creates a significant challenge for the convergence analysis of numerical schemes. To address this issue, we introduce novel error decomposition formulas and utilize higher-order consistency analysis. These techniques allow us to establish the ℓ∞ bound of the numerical solution under certain natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in ℓ∞(0,T;Hh−1)∩ℓ2(0,T;ℓ2). Furthermore, we conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes. These experiments include convergence tests and the observation of long-term coarsening dynamics, thereby confirming the robustness and reliability of our approach.

The energetics and colour for linearised models of wall turbulence

By comparing the budget of a data-driven quasi-linear approximation (DQLA) (Holford, Lee &amp; Hwang, J. Fluid Mech., vol. 980, 2024, A12) and direct numerical simulation (DNS) (Lee &amp; Moser, J. Fluid Mech., vol. 860, 2019, pp. 886–938), the energetics of linear models for wall-bounded turbulence are assessed. The DQLA is implemented with the linearised Navier–Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The self-consistent nature of the DQLA allows for a global comparison across all wavenumbers to assess the role of the various terms in the linear model in replicating the features present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a spectral budget equation is derived, connecting Lyapunov-like equations to the transport budget equations obtained from DNS. It is found that the DQLA and DNS are in good qualitative agreement for the streamwise-elongated structures present in DNS, comparing well for production, viscous transport and wall-normal turbulent transport. However, the DQLA does not have an energy-conservative nonlinear term. This results in no dissipation under molecular viscosity, with energy instead being dissipated locally through the eddy viscosity model, which models the energy removal by the nonlinear term at integral length scales. Comparison of the pressure–strain statistics also highlights the absence of the streak instability, with production and forcing mainly being retained in the streamwise and wall-normal components or shifted to the spanwise component. It is demonstrated that the eddy viscosity diffusion term locally enforces a self-similar budget, making the model for the nonlinear term self-consistent with a logarithmic mean profile. Implications and recommendations to improve the current eddy viscosity enhanced linear models are also discussed concerning the comparison with DNS, as well as considerations with regard to pressure statistics to mimic the role of the streak instability through colour of turbulence models.