Steady-state Diffusion Problems Research Articles

A 3D face interpolated discretisation method for simulating anisotropic diffusive processes on meshes coming from wood morphology

In this paper, a new 3D numerical discretisation method for solving the anisotropic, steady-state diffusion problem is developed and analysed. The scheme is constructed on hexahedral meshes using the geometrical properties of the cells. Indeed, each cell provides a local basis formed using the centres of its lateral faces. The discrete cell gradient approximation is then obtained by using three discrete directional derivatives resolved in terms of this basis, and by invoking the consistent relationships between opposite faces of the same cell. The face degrees of freedom are interpolated to reduce the complexity of the numerical scheme, resulting in the main unknowns being entirely nodal based. The scheme is unconditionally coercive and admits a unique solution. Various numerical experiments are performed to highlight the accuracy and the robustness of the method with respect to the mesh and anisotropy. An important outcome is that second order convergence is observed for all problems considered, even for highly deformed meshes. After this validation process, the method is applied to the prediction of the effective thermal conductivity of wood from its real 3D morphology. The property is estimated in the radial tangential and longitudinal directions. The solver is robust, efficient and yields to similar results compared to recent contributions.

On numerical solution stability of the Laplace equation in absence of the Dirichlet boundary condition: Benchmark problem proposal

A fundamental 1D steady state diffusion problem using only third type boundary condition also named as the Robin or Fourier type is analysed. In the heat transfer problems this boundary condition is called convective boundary condition using heat transfer coefficient ( α ) as a constant. The main idea of the proposed Benchmark problem is to test the solution accuracy in both limiting cases of α approaching to zero and to infinity. The infinite case limits to Dirichlet boundary condition and should not be the problem, while the zero limit leads to trivial solution of zeros in the absence of the Dirichlet boundary condition. The application of the Benchmark would be in freeze drying, where similar boundary conditions occur. The presented BEM numerical solution failed for α > 1 0 13 in double precision computation for two 2D formulations: cartesian and axisymmetric. The low limit results failed for α < 1 0 − 13 for cartesian formulation. This is expected, while the failure of axisymmetric formulation results already at moderate low α < 1 0 − 3 value is not. The solution stability and accuracy are found strong dependent to the fundamental solution integrals accuracy. The effect of system matrix preconditioner is also studied.

A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle

A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle

A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method

Physics informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. Here, the training of the network is equivalent to the minimization of a loss function that includes the governing (partial differential) equations (PDE) as well as initial and boundary conditions. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form (given governing equation) which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem (which can be obtained from the weak form) is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids and check the performance of the proposed PINN model against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). It is concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and data from physical FE simulations for a fast and accurate design of composite materials in future developments.

A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.

An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals

Abstract We present the first a priori error analysis of a new method proposed in Cockburn &amp; Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $u$ is the solution of a steady-state diffusion problem, the standard approximation $J(u_h)$ converges with order $h^{2k+1}$, where $u_h$ is the Hybridizable Discontinuous Galerkin approximation to $u$ with polynomials of degree $k&amp;gt;0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $h^{4k}$, and can be computed by only using twice the computational effort needed to compute $J(u_h)$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $u$ or the functional $J(\cdot )$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods.

Modeling the Age-Hardening Process of Aluminum Alloys Containing the Prolate/Oblate Shape Precipitates

Non-spherical precipitates are the main strengthening source of the age-hardenable aluminum alloys. In the majority of the precipitation hardening models presented so far, the simple spherical shape has been assumed. Moreover, the models which considered the actual shape of the precipitates, are derived based on the simple assumption of steady-state diffusion problem solutions. In the present study, the classical Kampmann and Wagner numerical model of spherical precipitates is extended to model the kinetics of spheroidal shape precipitates evolution during the aging treatment of Al alloys. To do so, a new rate law is proposed using a similarity solution of the transient diffusion problem around the spheroidal precipitates, with different aspect ratios, and moving boundaries. Moreover, a modified age-hardening model which considered the effects of precipitate shape, size and volume fraction, is used to predict the variation of the alloy hardness during the aging process. The accuracy of the proposed model is shown by comparing the predicted features of precipitates, and hardness evolution with the published experimental data. Also, the validity of the existing approximate solutions for the aging problem is discussed.

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.

Microscopic Models of Drug/Chemical Diffusion Through the Skin Barrier: Effects of Diffusional Anisotropy of the Intercellular Lipid

Owing to the systematic alignment and ordering of fatty acid and ceramide chains, lipid layers in biological membranes have strongly anisotropic diffusion properties. The diffusivity D∥lip for solute transport in the direction parallel to the lipid layer is typically 103-105 times the diffusivity D⊥lip for the perpendicular direction. This article explores the consequences of this strong degree of anisotropy on solute diffusion through the stratum corneum (barrier) layer of the skin based on a realistic representation of a unit cell of the microstructure. Complementary numerical methods (smoothed particle hydrodynamics, finite differences) are used to solve the steady-state unit-cell diffusion problem leading to the average (homogenized, coarse-grained) diffusion tensor characterizing the tissue as an effective continuum. A parametric study is presented characterizing solute concentration profiles in detail for testosterone- and caffeine-like permeants, and it is shown that the results cannot be mimicked by calculations based on an isotropic lipid-phase diffusivity. The ratio of lateral to transdermal effective diffusivities calculated by the present model is of the order of 40 and 300 for fully hydrated (in vitro) and partially hydrated (in vivo) states, respectively. These values compare favorably with the results of recent experiments.

Two algorithms for fast 2D node generation: Application to RBF meshless discretization of diffusion problems and image halftoning

Mesh generation techniques for traditional mesh based numerical approaches such as FEM and FVM have now reached a good degree of maturity. There is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches, despite their theoretical simplicity and efficiency; furthermore node generation can be put in connection with some well-known image approximation techniques. Two node generation algorithms are here proposed and employed in the numerical solution of 2D steady state diffusion problems by means of a local Radial Basis Function (RBF) meshless method. Finally, such algorithms are also tested for grayscale image approximation through stippling.

A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The method is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

Galerkin based generalized ANOVA for the solution of stochastic steady state diffusion problems

This work presents a novel approach, referred here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA), for the solution of stochastic steady state diffusion problems. The proposed approach utilizes generalized ANOVA (G-ANOVA) expansion to represent the unknown stochastic response and Galerkin projection to decompose the stochastic differential equation into a set of coupled differential equations. The coupled set of partial differential equations obtained are solved using finite difference method and homotopy algorithm. Implementation of the proposed approach for solving stochastic steady state diffusion problems has been illustrated with three numerical examples. For all the examples, results obtained are in excellent agreement with the benchmark solutions. Additionally, for the second and third problems, results obtained have also been compared with those obtained using polynomial chaos expansion (PCE) and conventional G-ANOVA. It is observed that the proposed approach yields highly accurate result outperforming both PCE and G-ANOVA. Moreover, computational time required using GG-ANOVA is in close proximity of G-ANOVA and less as compared to PCE.

The boundary element method applied to the solution of two-dimensional diffusion–advection problems for non-isotropic materials

A boundary element method (BEM) formulation is developed for the analysis, in the time-domain, of the diffusion–advection problem for non-isotropic materials in two dimensions. As the diffusion–advection equation describes, among others, the pollution dispersion problem, the development of formulations capable of dealing with this social and environmental problem is always welcome. The formulation presented here employs the fundamental solution of the steady-state pure diffusion problem. Therefore, the resulting BEM equation presents three domain integrals, related to the velocity components, to the decay term, and to the time-derivative of the variable of interest. This time-derivative is approximated by means of a backward finite difference scheme. Three examples are presented and discussed at the end of the article. The first example shows how varying the value of the diffusion coefficient in one direction, while keeping a constant value in the other direction, influences the results in a pure diffusion problem. The second example deals with a pollutant dispersion problem, and consequently, is described by the diffusion–advection equation. The third example also deals with a pollutant dispersion problem, but the analysis is carried out in a linear channel with a variable width.

Non-local model for diffusion-mediated dislocation climb and cavity growth

To design efficient thermal recovery procedures for structural materials in fusion energy applications it is important to characterise quantitatively the annealing timescales of radiation-induced defect clusters. With this goal in mind, we present an extension of the Green’s function formulation of Gu et al. (2015). for the climb of curved dislocations, to include in the same framework the evaporation and growth of cavities and the effects of free surfaces. This paper focuses on the mathematical foundations of the model, which makes use of boundary integral equations (París and Cañas, 1997) to solve the steady-state vacancy diffusion problem. Numerical results are also presented in the simplified case of a dilute configuration of prismatic dislocation loops and spherical cavities in a finite-size medium, which show good agreement with experimental data on high temperature annealing in ion-irradiated tungsten (Ferroni et al., 2015).

Superconvergence byM-decompositions. Part III: Construction of three-dimensional finite elements

We apply the concept of an M -decomposition in the framework of steady-state diffusion problems to construct local spaces defining superconvergent hybridizable discontinuous Galerkin methods as well as their companion sandwiching mixed methods in ℝ 3 with tetrahedral, pyramidal, prismatic, and hexahedral elements.

Superconvergence byM-decompositions. Part II: Construction of two-dimensional finite elements

We apply the concept of an M -decomposition introduced in Part I to systematically construct local spaces defining superconvergent hybridizable discontinuous Galerkin methods, and their companion sandwiching mixed methods. This is done in the framework of steady-state diffusion problems for the h - and p -versions of the methods for general polygonal meshes in two-space dimensions.

Modeling diffusion of sulfate through concrete using mixture theory

Concrete exposed to sulfates deteriorates. Here, an attempt is made to see whether the framework of mixture theory can be used to model the changes that occur in concrete exposed to sodium sulfate. Toward this, diffusion and reaction of sulfates with concrete is modeled within the framework of mixture theory. Appropriate choices are made for the Helmholtz free energy and interaction momentum so that the process of diffusion of sulfates in concrete can be captured. As expected in mixture theory, diffusion causes deformation of the solid. The parameters in the mixture theory model are determined by comparing the steady-state concentration profile of the diffusing sodium sulfate solution and inlet velocity of the fluid with that of Fick’s solution for the same boundary conditions. An assumption is made for the mass production term to capture the reaction of sulfates with concrete. The rate constant and order of the reaction are estimated using the concentration of gypsum, calcium hydroxide and ettringite reported in the literature, for various duration of exposure of cement pastes to known concentration of sodium sulfate solution. Finally, the governing equations for the combined problem of steady-state diffusion and reaction of sulfates with concrete are presented. A numerical scheme to solve the governing equations is outlined. Long-term concentration profiles of sodium sulfate predicted by the framework agree qualitatively with the experimentally observed profile reported in the literature.

Amplification of salt-induced polymer diffusiophoresis by increasing salting-out strength.

The role of salting-out strength on (1) polymer diffusiophoresis from high to low salt concentration, and (2) salt osmotic diffusion from high to low polymer concentration was investigated. These two cross-diffusion phenomena were experimentally characterized by Rayleigh interferometry at 25 °C. Specifically, we report ternary diffusion coefficients for polyethylene glycol (molecular weight, 20 kg·mol(-1)) in aqueous solutions of several salts (NaCl, KCl, NH4Cl, CaCl2, and Na2SO4) as a function of salt concentration at low polymer concentration (0.5% w/w). We also measured polymer diffusion coefficients by dynamic light scattering in order to discuss the interpretation of these transport coefficients in the presence of cross-diffusion effects. Our cross-diffusion results, primarily those on salt osmotic diffusion, were utilized to extract N(w), the number of water molecules in thermodynamic excess around a macromolecule. This preferential-hydration parameter characterizes the salting-out strength of the employed salt. For chloride salts, changing cation has a small effect on N(w). However, replacing NaCl with Na2SO4 (i.e., changing anion) leads to a 3-fold increase in N(w), in agreement with cation and anion Hofmeister series. Theoretical arguments show that polymer diffusiophoresis is directly proportional to the difference N(w) - n(w), where n(w) is the number of water molecules transported by the migrating macromolecule. Interestingly, the experimental ratio, n(w)/N(w), was found to be approximately the same for all investigated salts. Thus, the magnitude of polymer diffusiophoresis is also proportional to salting-out strength as described by N(w). A basic hydrodynamic model was examined in order to gain physical insight on the role of n(w) in particle diffusiophoresis and explain the observed invariance of n(w)/N(w). Finally, we consider a steady-state diffusion problem to show that concentration gradients of strong salting-out agents such as Na2SO4 can produce large amplifications and depletions of macromolecule concentration. These effects may be exploited in self-assembly and adsorption processes.

Optimized Schwarz methods for a diffusion problem with discontinuous coefficient

We study non-overlapping Schwarz methods for solving a steady-state diffusion problem in heterogeneous media. Various optimized transmission conditions are determined by solving the corresponding min-max problems; we consider different choices of Robin conditions and second order conditions. To compare the resulting methods, we analyze the convergence in two separate asymptotic regimes: when the mesh size is small, and when the jump in the coefficient is large. It is shown that optimized two-sided Robin transmission conditions are very effective in both regimes; in particular they give mesh independent convergence. Numerical experiments are presented to illustrate and confirm the theoretical results.

Piezoresistive device optimization using topological derivative concepts

A piezoresistive sensor is composed of a piezoresistive membrane attached to a flexible plate. The piezoresistive material is anisotropic, and its electrical properties change when subjected to mechanical stresses. In this work, the topology design of a piezoresistive pressure sensor is addressed. More specifically, an optimization technique based on topological sensitivity analysis is proposed in order to obtain the optimized distribution of piezoresistive material over the plate. In most of the works regarding topological sensitivity analysis, isotropic materials are considered. However, the problem of conductivity in an anisotropic non-homogeneous domain has been recently addressed, and a closed form for the topological derivative associated to the energy shape functional has been presented. In this work, on the other hand, a closed form for the topological derivative associated with a multi-objective shape functional related to the steady-state anisotropic current density diffusion problem is presented. To illustrate the applicability of the closed formula and the proposed optimization procedure, numerical examples regarding the conceptual design of piezoresistive sensors, considering distinct optimization parameters and boundary conditions in the conductivity problem, are presented.