One-dimensional Diffusion Problem Research Articles

Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models are integral equations that are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and the nonlocality disappears as the interaction radius (horizon) vanishes, then the ND problem recovers the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may contain discontinuities, and this is one of the properties of the ND problem that sets it apart from the classic diffusion problem. It is natural to adopt the DG method to compute the ND problems since the DG method shows its great advantages in resolving problems with discontinuities in the computational fluid dynamics in the past several decades. Based on \cite{DuCAMC2020}, we construct the DG methods with different penalty terms, and the proposed DG methods have their local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial to obtain the {\it asymptotic compatibility}. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To see the nonlocal diffusion effect, we also consider the time-dependent convection-diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers' equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments.

Stabilization of a thermoelastic diffusion system of type II damped and delayed

ABSTRACT The present paper is devoted to the study of exponential stabilization of a one-dimensional thermoelastic diffusion problem of type II for isotropic and homogeneous materials with frictional damping and time delay. The system of equations under consideration is a coupling of three hyperbolic equations involving three variables considering frictional damping and time delay on each variable. Using semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model through the Gearhart–Huang–Pruss theorem.

Model reduction for molecular diffusion in nanoporous media

Porous materials are widely used for applications in gas storage and separation. The diffusive properties of a variety of gases in porous media can be modeled using molecular dynamics simulations that can be computationally demanding depending on the pore geometry, complexity, and amount of gas adsorbed. We explore a dimensionality reduction approach for estimating the self-diffusion coefficient of gases in simple pores using Langevin dynamics, such that the three-dimensional (3D) atomistic interactions that determine the diffusion properties of realistic systems can be reduced to an effective one-dimensional (1D) diffusion problem along the pore axis. We demonstrate the approach by modeling the transport of nitrogen molecules in single-walled carbon nanotubes of different radii, showing that 1D Langevin models can be parametrized with a few single-particle 3D atomistic simulations. The reduced 1D model predicts accurate diffusion coefficients over a broad range of temperatures and gas densities. Our work paves the way for studying the diffusion process of more general porous materials such as zeolites or metal-organics frameworks with effective models of reduced complexity.

ФУНКЦИЯ НАДЕЖНОСТИ ПРЕДВАРИТЕЛЬНО НАПРЯЖЕННОЙ КОРРОДИРОВАННОЙ ЖЕЛЕЗОБЕТОННОЙ БАЛКИ ПРИ НЕЛИНЕЙНОМ РАСПРОСТРАНЕНИИ КОРРОЗИИ

The paper considers the limit state function in the context of reliability analysis of corroded beams. Methods for determining the ultimate load limit for corroded beams are discussed. Various approaches to the determination of the limit state function are considered, including those based on statistical data on corrosion and modelling of stresses in a reinforced concrete beam. A general reliability assessment model for reinforced concrete structures should include corrosion propagation. Most previous studies have focused on one-dimensional diffusion problems with an assumed constant corrosion rate. A nonlinear corrosion rate model, unlike linear models, considers the corrosion current density not constant over the service life of a reinforced concrete structure. An approach to reliability analysis together with a nonlinear corrosion growth model is developed. In this paper, the main problems related to reliability under a nonlinear model of prestressed reinforcement rope corrosion of reinforced concrete beams are discussed. A function for the reduction of reinforcing bar diameter from time is presented. Sensitivity analysis is performed to determine the effect of corrosion growth parameters on the reliability index of reinforced concrete T-beam. The nonlinear corrosion growth model, together with other relevant probabilistic models used to describe random variables, was applied to analyze the reliability of a reinforced concrete crane girder. The expression of corrosion current density shows that the corrosion rate increases exponentially as the value of the design parameter of the model increases. To further evaluate the effect of the proposed time corrosion growth model on the reliability of reinforced concrete beam, two specific cases are considered. The first one is with fixed corrosion current density, and the second one is with fixed corrosion growth at a given time.

Diffusion of light in turbid media with internal reflections

We relate the the Kubelka-Munk equations for the description of the intensity transfer of light in turbid media to a one-dimensional diffusion equation, which is obtained by averaging the three-dimensional diffusion equation over the lateral directions. This enables us to identify uniquely the Kubelka-Munk parameters and derive expressions for diffuse reflection and transmission coefficients including the effect of internal reflections. Without internal reflections we recover the Kubelka-Munk formulae for these coefficients. We show that the Kubelka-Munk equations are the proper radiative-transfer equations for the one-dimensional diffusion problem.

The finite element model of chloride diffusion in pre-stressed corroded reinforcement bars of reinforced concrete structures

Introduction. The general corrosion model, made for reinforced concrete structures, must include the initiation of environmental influences such as carbonation, cracking and chloride ion penetration. It depends on the rate and degree of corrosion, corrosive effects in reduced areas, as well as the lower strength of bond between pre-stressed and unstressed reinforcement bars and concrete. The majority of earlier studies were focused on one-dimensional diffusion problems with an assumed constant corrosion rate.
 
 Materials and methods. In the aftermath of general corrosion, localized corrosion is accompanied by a release of hydrogen and alkaline water, chlorine ions. Crack propagation in reinforcement wires is calculated using the transient finite element software for chloride diffusion, which is time-dependent.
 
 Results. In most cases, the diffusion equation does not have a closed form solution, and therefore, the finite difference method can be used. The authors have shown that the corrosion rate decreases with time if current density has different water-cement ratios. If the water-cement ratio increases, the corrosion rate increases, as well.
 
 Conclusions. In a pre-stressed reinforced concrete beam, the corrosion of one bar affects the total corrosion of all bars and the reduction in the cross-sectional area of bars that does not exceed 15 %.

Quantitative imaging and modeling of colloidal gelation in the coagulant dipping process.

Many common elastomeric products, including nitrile gloves, are manufactured by coagulant dipping. This process involves the destabilization and gelation of a latex dispersion by an ionic coagulant. Despite widespread application, the physical chemistry governing coagulant dipping is poorly understood. It is unclear which properties of an electrolyte determine its efficacy as a coagulant and which phenomena control the growth of the gel. Here, a novel experimental protocol is developed to directly observe coagulant gelation by light microscopy. Gel growth is imaged and quantified for a variety of coagulants and compared to macroscopic dipping experiments mimicking the industrial process. When the coagulant is abundant, gels grow with a t1/2 time dependence, suggesting that this phenomenon is diffusion-dominated. When there is a finite amount of coagulant, gels grow to a limiting thickness. Both these situations are modeled as one-dimensional diffusion problems, reproducing the qualitative features of the experiments including which electrolytes cause rapid growth of thick gels. We propose that the gel thickness is limited by the amount of coagulant available, and the growth is, therefore, unbounded when the coagulant is abundant. The rate of the gel growth is controlled by a combination of a diffusion coefficient and the ratio of the critical coagulation concentration to the amount of coagulant present, which in many situations is set by the coagulant solubility. Other phenomena, including diffusiophoresis, may make a more minor contribution to the rate of gel growth.

Development of the Established Redlich-Kister Finite Difference Solution with MKSOR Iteration for Solving One Dimensional Diffusion Problems

This paper contributes a new method known as the second-order Redlich-Kister Finite Difference (RKFD) solution to the partial differential equation, especially for one-dimensional (1D) diffusion problems. All derivative terms for the proposed method are needed for the discretization process of the second-order RKFD. Arranging the derivative terms will lead to the second-order RKFD approximation equations. Then, this approximation equation is applied to solve the system of the RKFD equation. As the large-scale and sparse coefficient matrix is obtained, it will be solved iteratively to regulate the high computational complexity by using the Gauss-Seidel (GS), Kaudd Successive Over Relaxation (KSOR) and Modified Kaudd Successive Over Relaxation (MKSOR) iterative methods. All of those iterative methods are developed according to the matrix structure of the system and are applied to three examples of the proposed problem. As a result, MKSOR iterative method showed significant improvement in terms of performance efficiency by contrast to GS and KSOR iterative methods. The performance efficiency is measured by the number of iterations, execution time and maximum norm.

A Comparative Study of Fatigue Energy Dissipation of Additive Manufactured and Cast AlSi10Mg Alloy

In this paper, the fatigue energy dissipation of Gravity Casting (GC) and Laser-based Powder Bed Fusion (LPBF) AlSi10Mg alloys under cyclic loading are investigated. The increase in surface temperature related to the energy dissipation effect is decoupled and used to predict the fatigue limits of GC and LPBF AlSi10Mg alloys as being 55.8% UTS and 33.9% UTS, respectively. The energy dissipation rate is obtained by solving the one-dimensional thermal diffusion problem. This energy dissipation is separated into related and unrelated fatigue damage using polynomial function fitting. The energy dissipation related to fatigue damage for LPBF specimens is observed to be higher than that of GC specimens, which indicates worse fatigue performance. The fatigue damage entropy is employed to predict the fatigue life of both GC and LPBF AlSi10Mg alloys, which has a good agreement with the results of a traditional fatigue experiment.

Imaginary Eigenvalues in Multilayer One-Dimensional Thermal Conduction Problem with Linear Temperature-Dependent Heat Generation

Diffusion in a multi-layer body is a common problem in heat and mass transfer, with applications in multiple engineering systems, such as Li-ion battery packs and first-order chemical reactions occurring in a multilayer body. Development of analytical models to describe diffusion in such systems is helpful for both heat and mass transfer. This paper addresses a multi-layer one-dimensional diffusion problem, in which, generation/consumption in each layer is proportional to the local temperature/concentration. This could occur, for example, due to temperature-dependent heat generation or species generation/consumption associated with a first-order chemical reaction. It is shown that eigenvalues of this problem may become imaginary under two distinct conditions. A physical interpretation of these conditions is discussed, and a mathematical requirement for existence of imaginary eigenvalues is derived. The relationships between imaginary eigenvalues and various non-dimensional problem parameters are discussed. It is also shown that the computed temperature in the multilayer body remains real even if some eigenvalues may become imaginary. Therefore, all eigenvalues, whether real or imaginary must be accounted for in temperature computation. While presented in the context of heat transfer, these results are also valid for multi-layer mass transfer problems involving species generation/consumption due to chemical reaction. This work improves the theoretical understanding of diffusion in a multilayer body under conditions relevant for several engineering processes and systems.

Polydisperse Colloids Two-Moment Diffusion Model Through Irreversible Thermodynamics Considerations

Abstract This study deals with the problem of diffusion for polydisperse colloids. The resolution of this complex problem usually requires computationally expensive numerical models. By considering the number of colloidal particles and their mass as independent variables, the equations of state for a dilute polydisperse colloid are derived on a statistical mechanics basis. Irreversible thermodynamics is then applied to obtain a simple two-moment diffusion model. The validity of the model is illustrated by comparing its results with those obtained by a classical size spectrum approach, in a sedimentation equilibrium problem and in an unsteady one-dimensional diffusion problem in Stokes–Einstein regime, and under the hypothesis that the size spectrum distribution is stochastic. In the first problem, the two-moment diffusion problem allows to represent rigorously the vertical size segregation induced by gravity, while in the second one, it allows a convenient description of the diffusion of polydisperse colloids by using two coupled diffusion equations, with an accuracy comparable with that of the classical size spectrum approach. The contribution of our work lies primarily in the application of a non-equilibrium thermodynamics methodology to a challenging issue of colloid modeling, namely, polydispersity, by going from statistical mechanics to the derivation of phenomenological coefficients, with the two-moment approach as a guideline.

Predictions of solute mixing in a weld pool and macrosegregation formation during dissimilar-filler welding of aluminum alloys: Modeling and experiments

The solute mixing phenomena of filler and base metal melts during dissimilar-filler metal-inert gas (MIG) welding are simulated by a multi-phase lattice Boltzmann (LB) model for predicting macrosegregation formation in the weld joint. The LB models incorporates the calculations of fluid field and solute transport, and the quantitative capalibity is validated through the problem of one-dimensional liquid diffusion. For the MIG welding of AA6063 sheets with an ER5183 filler wire, the simulation results reveal that the periodic impingement of heterogeneous filler metal droplets leads to symmetrical vortex flows accompanied by the unmixed zones formed in the center. These unmixed zones locate in the bulk weld pool and would develop into macrosegregation in the weld joint. In addition, the fluid flow near the fusion boundary remains almost stagnant, which gives rise to the appearance of the thereby continuous unmixed zone. The predicted distribution of unmixed zone in the weld pool agree well with the macrosegregation patterns in the weld joint observed in the experiments. For partially mimicking the effect of external magnetic field on droplet movement recorded by a high-speed camera, random droplet flight directions are arranged in the LB simulations. With the increase of droplet impact speed (Weber number) and the arrangement of random droplet flight direction, the simulated Mg concentration in the weld pool becomes more uniform, implying that the risk of macrosegregation formation is reduced. The LB simulations provides insight for better understanding the formation of weld defects during MIG welding with a dissimilar filler metal.

Boundary Stabilization of a Thermoelastic Diffusion System of Type II

In this paper we study the boundary stabilization of a one-dimensional thermoelastic diffusion problem of type II. The system of equations is a coupling of three hyperbolic equations. This poses some new mathematical and numerical difficulties. With the help of the semigroup theory of linear operators, we prove the well-posedness of the proposed problem. By using the frequency domain method combined with the multiplier technique, we prove the exponential stability of the solutions. Finally, we present a numerical scheme based on the Chebyshev spectral method and we give two numerical examples to validate the proposed model and to show its capability.

Data assimilation method to de-noise and de-filter particle image velocimetry data

We present a variational data assimilation method in order to improve the accuracy of velocity fields $\tilde{\boldsymbol{v}}$, that are measured using particle image velocimetry (PIV). The method minimises the space–time integral of the difference between the reconstruction $\boldsymbol{u}$ and $\tilde{\boldsymbol{v}}$, under the constraint, that $\boldsymbol{u}$ satisfies conservation of mass and momentum. We apply the method to synthetic velocimetry data, in a two-dimensional turbulent flow, where realistic PIV noise is generated by computationally mimicking the PIV measurement process. The method performs optimally when the assimilation integration time is of the order of the flow correlation time. We interpret these results by comparing them to one-dimensional diffusion and advection problems, for which we derive analytical expressions for the reconstruction error.

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems

There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local diffusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the effectiveness and the robustness of the proposed method.

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper, we discuss the local discontinuous Galerkin methodscoupled with two specific explicit-implicit-null time discretizations forsolving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$.The basic idea is to add and subtract two equal terms $a_0~U_{xx}$on the right-hand side of the partial differential equation,then to treat the term $a_0~U_{xx}$ implicitly and the other terms$(a(U)U_x)_x-a_0~U_{xx}$ explicitly. We give stability analysis for themethod on a simplified model by the aid of energy analysis,which gives a guidance for the choice of $a_0$, i.e., $a_0~\ge~\max\{a(u)\}/2$to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model, andnumerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.

On the Possibility of a Numerical Solution of the Heat and Mass Transfer Problem with the Combined Matrix&Generalized Powers of Bers Method

Some possibilities for the combined use of the matrix method and the method of generalized powers of Bers for the numerical solution of the heat and mass transfer problem is considered. Initially, the matrix method under consideration was used by us for the analytical solution of the problem of heat and mass transfer in a homogeneous or multilayered medium with translation, axial or central symmetry for an arbitrary number of layers. Simulation is reduced to the sequential multiplication of second-order functional matrices whose components at each point are determined by the physical and geometric parameters of the current layer. The main advantage of the matrix method is that it can be used for any number of layers. It is this property of the method that makes it possible to consider of its application as a numerical one if the medium is artificially splitted into many layers.For modelling, the problem of one-dimensional diffusion of minority charge carriers generated by a wide electron beam in a semiconductor material was taken. The relative error in the computations was estimated by a uniform norm in comparison with the analytical solution. The errors made were from 0.8% to 2.9%.

Simulink Model for PWM-Supplied Laminated Magnetic Cores Including Hysteresis, Eddy-Current, and Excess Losses

A new implementation of an iron-loss model for laminated magnetic cores in the MATLAB/Simulink environment is proposed in this paper. The model is based on numerically solving a one-dimensional diffusion problem for the eddy currents in the core lamination and applying an accurate hysteresis model as the constitutive law. An excess loss model is also considered. The model is identified merely based on the catalog data provided by the core material manufacturer. The implementation is validated with analytical and finite-element models and experimentally in the case of a toroidal inductor supplied from a GaN FET full-bridge inverter with 5–500 kHz switching frequencies and a deadtime of 300 ns. Despite the simple identification, a good correspondence is observed between the simulated and measured iron losses, the average difference being 3.3% over the wide switching frequency range. It is shown that accounting for the skin effect in the laminations is significant, in order to correctly model the iron losses at different switching frequencies. Some differences between the measured and simulated results at high switching frequencies are also discussed. The model is concluded to be applicable for designing and analyzing laminated magnetic cores in combination with power-electronics circuits. The Simulink models are openly available.