Nonlinear Diffusion Problems Research Articles

Modeling nonlinear moisture diffusion in inhomogeneous media

While moisture diffusion in microelectronic device and packaging has been studied for decades, the problems involving complex nonlinear moisture diffusion in multi-material assembly have not been fully studied. This paper has developed a general nonlinear diffusion model by adopting water activity, a continuous state variable, as the field variable. The generalized solubility is introduced, which is temperature- and water activity-dependent. The effective diffusivity is defined and derived in terms of generalized solubility and water activity. By comparing the water activity-based model with the existing various normalized models, the present theory can unify and generalize the current approaches. More importantly, the present model can solve both linear and nonlinear moisture diffusion in inhomogeneous material system without normalization. The commercial finite element software has been applied to solve the nonlinear generalized moisture diffusion problem using the analogy of water activity and temperature. A source code of user-defined subroutines in ABAQUS has been provided in the Appendix of the paper. The mathematical formulation and the numerical implementation method presented in this paper can be applied to any nonlinear sorption or diffusion problems in inhomogeneous material system.

Modelling of gas flow in shale using a finite volume method

• The paper tries to explain a gas shale flow model more clearly and rigorously. • A finite volume method for solving the gas shale flow equation was developed. • Inversion is performed for three flow parameters. Gas flow in shale is a very complex phenomenon, currently investigated using a variety of techniques including the analysis of transient experiments conducted on full core and crushed shale using a range of gases. A range of gas flow mechanisms may operate in shale including continuum flow, slippage, transitional flow and Knudsen diffusion. These processes, as well as gas sorption, need to be taken into account when interpreting experimental data and extrapolating the results to the subsurface. Several models have been published that attempt to account for these different processes. Unfortunately, these have a large number of unknown parameters and few studies have assessed the extent to which transient experiments may be used to invert for the key unknowns or the errors that are associated. Here we present a methodology in which various inversion techniques are applied to assess the viability of deriving key unknowns which control gas flow in shale from transient experiments with a range of noise. A finite volume method is developed for solving the model of Civan (2010, 2011a,b) of transient gas flow in shale. The model is applicable to non-linear diffusion problems, in which the permeability and fluid density both depend on the scalar variable, pressure. The governing equation incorporates the Knudsen number, allowing different flow mechanisms to be addressed, as well as the gas adsorption isotherm. The method is verified for unsteady-state problems for which analytical or numerical solutions are available. The method is then applied to a pressure-pulse decay test. An inverse numerical formulation is generated, using a minimization iterative algorithm, to estimate some unknown physical parameters. Both numerically simulated noisy and experimental data are input into the formulation of the inverse problem. Error analysis is undertaken to investigate the accuracy of results. A good agreement between inverted and exact parameter values is obtained for several parameters. However, it was found that the strong correlation between intrinsic permeability and tortuosity meant that it was not possible to accurately invert simultaneously for these two parameters from the current pressure-pulse decay model.

Estimation of the non-linear diffusion coefficient with Marcov Chain Monte Carlo method based on the integral information

PurposeThis paper aims to present development and application of the Bayesian inverse approach for retrieving parameters of non-linear diffusion coefficient based on the integral information.Design/methodology/approachThe Bayes formula was used to construct posterior distribution of the unknown parameters of non-linear diffusion coefficient. The resulting aposteriori distribution of sought parameters was integrated using Markov Chain Monte Carlo method to obtain expected values of estimated diffusivity parameters as well as their confidence intervals. Unsteady non-linear diffusion equation was discretised with the Global Radial Basis Function Collocation method and solved in time using Crank–Nicholson technique.FindingsA number of manufactured analytical solutions of the non-linear diffusion problem was used to verify accuracy of the developed inverse approach. Reasonably good agreement, even for highly correlated parameters, was obtained. Therefore, the technique was used to compute concentration dependent diffusion coefficient of water in paper.Originality/valueAn original inverse technique, which couples efficiently meshless solution of the diffusion problem with the Bayesian inverse methodology, is presented in the paper. This methodology was extensively verified and applied to the real-life problem.

Numerical Methods for Power-Law Diffusion Problems

In this paper, we consider numerical methods for nonlinear diffusion problems where the diffusion term follows a power law, e.g., $p$-Laplace-type problems. In the first part, we present continuous higher order finite element discretizations for the model problem and we derive error estimates. In the second part, we discuss Newton iterative methods based on residual-based line-search and error-oriented globalization, which are employed for the numerical solution of the produced nonlinear algebraic system. Third, we formulate the original problem as a saddle point problem in the frame of augmented Lagrangian techniques and present two iterative methods for its solution. We conduct a systematic investigation of all solution algorithms. These algorithms are compared with respect to computational cost and their efficiency. Numerical results demonstrating the theoretical error estimates are also presented in five examples.

Coupled Hydro-Mechanical Model of Bentonite Hydration and Swelling

This paper deals with the modelling of coupled hydro-mechanical processes at the buffer and host rock interface (bentonite and granite) in the context of the safe disposal of spent nuclear fuel. Granite, as one of the barriers, includes fractures which are the source for hydration of bentonite and its subsequent swelling. It affects the mechanical behaviour and possibly the stability of the whole system. A non-linear solution for the stress-deformation problem with swelling was developed. This solution is coupled with the non-linear diffusion problem (for unsaturated flow). The swelling is defined using a coefficient dependent on water content according to literature data, with the effective Young's modulus decreasing close to zero corresponding to the plastic state. Results confirm the expected non-uniform saturation, swelling, and stresses in bentonite and small contribution to a fracture displacement.

Improved Method for Determining the <inline-formula> <tex-math notation="LaTeX">$n$</tex-math> </inline-formula>-Value of HTS Bulks

The complete penetration magnetic field $B_{P}$ is the main feature of a superconducting pellet submitted to an axial applied magnetic field. The electric $E{-}J$ characteristics of HTS bulk is generally described by a power law $E(J) = E_{C}(J/J_{C})^{n}$ . The influence of the $n$ -value and applied magnetic field rise rate $V_{b}$ on the $B_{P}$ of a HTS cylindrical pellet has been presented in a previous paper. The numerical results presented come from a numerical resolution of a nonlinear diffusion problem. With the help of these simulations, a linear relationship between $B_{P}$ , ln $V_{b}$ , and $n$ -value has been deduced. This comparison allows determining the critical current density $J_{C}$ and the $n$ -value of the power law based on direct measurement of $B_{P}$ in the gap between two bulk HTS pellets. In this paper, an improvement of this method is presented. The influence of geometric parameters $R$ and $L$ is studied to give generality to the relationship between $B_{P}$ , $V_{b}$ , and $n$ -value. A previous $B_{P}$ formula is confirmed by these new simulations. To correctly connect simulation and experimental results, the influence of spacing $e$ between bulks is studied and presented. A relationship between $B_{P}$ and measured complete penetration magnetic field $B_{PM}$ is determined.

3-D Modeling of Heterogeneous and Anisotropic Superconducting Media

In this paper, we investigate the 3-D computation of the current distribution in the heterogeneous superconducting media with a discontinuous anisotropic $E$ – $J$ constitutive power law. The resulting 3-D vectorial non-linear diffusion problem satisfied by the electric field is solved in each region of the media with a discontinuous Galerkin method combined with a 3-D semi-implicit scheme recently proposed for the 3-D modeling of homogeneous bulk superconductors. Numerical applications on the media consisting of two embedded anisotropic superconducting regions with and without a conductive matrix are computed. Both critical current density $J_{c}$ and power law exponent $n$ are uniform in each region but discontinuous in the media. The results will show the continuity of the diffusion phenomena in the media and the differences between both isotropic and anisotropic cases under an applied axial sinusoidal magnetic field.

Spreading and vanishing in nonlinear diffusion problems with free boundaries

We study nonlinear diffusion problems of the form u_t=u_{xx}+f(u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f(u) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f(u) which is C^1 and satisfies f(0)=0 , we show that the omega limit set \omega(u) of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter \sigma in the initial data, we reveal a threshold value \sigma^* such that spreading ( \lim_{t \to \infty}u= 1 ) happens when \sigma > \sigma^* , vanishing ( \lim_{t \to \infty}u=0 ) happens when \sigma < \sigma^* , and at the threshold value \sigma^* , \omega(u) is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.

Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem.

Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second‐order time evolution

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

SummaryA numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L2‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.

Application of the Method of Galerkin to Non Linear Problem Stochastic Heat Conduction One-Dimensional

The Galerkin method is applied in the non-linear stochastic diffusion problem. The uncertainty is present in the coefficients of diffusion equation. The uncertainty is modeled by random variables. The chaos polynomials is used to approximate the stochastic behavior of the problem. The approximate solutions obtained through Galerkin method are compared with Monte Carlo simulation in terms of the statistical moments of random variables generated by the random field solution.

Determination of <inline-formula> <tex-math notation="LaTeX">$J_C$</tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n$</tex-math></inline-formula>-Value of HTS Pellets by Measurement and Simulation of Magnetic Field Penetration

Complete penetration magnetic field BP is a feature of a superconducting sample submitted to an applied magnetic field. It is very important to know this for applications such as an electrical motor or levitation. The electric E-J characteristics of a high-temperature superconductor (HTS) bulk is generally described by a power law. The main purpose of this paper is to investigate the influence of the n-value and the applied magnetic field rise rate Vb on the BP of a cylindrical HTS pellet. The numerical results presented come from the resolution of a nonlinear diffusion problem with commercial software. In this paper, cylindrical HTS pellets are submitted to an axial applied magnetic field. With the help of these simulations, a linear relationship between BP , Vb, and the n-value has been found. A comparison between measurements and simulations is done for the magnetization of cylindrical bulk superconducting samples. This comparison allows to determine the critical current density J C and n-value of the power law E(J) = EC(J/J C )n. The experiment is based on the direct measurement of the local magnetic field in the gap between two bulk HTS pellets. The field penetration measurements have been carried out on HTS pellets at 77 K by applying increasing magnetic fields with a quasi-constant sweep rate for the axial direction of the applied magnetic field. Two values of complete penetration magnetic field BP have been measured at two different rise rates Vb. The n-value of the real HTS pellet has been deduced.

A Study on Anisotropic Mesh Adaptation for Finite Element Approximation of Eigenvalue Problems with Anisotropic Diffusion Operators

Anisotropic mesh adaptation is studied for the linear finite element solution of eigenvalue problems with anisotropic diffusion operators. The $\mathbb{M}$-uniform mesh approach is employed with which any nonuniform mesh is characterized mathematically as a uniform one in the metric specified by a metric tensor. Bounds on the error in the computed eigenvalues are established for quasi-$\mathbb{M}$-uniform meshes. Numerical examples arising from the Laplace--Beltrami operator on parameterized surfaces and nonlinear diffusion problems in image processing and radiation hydrodynamics are presented. Numerical results show that anisotropic adaptive meshes can lead to more accurate computed eigenvalues than uniform or isotropic adaptive meshes. They also confirm the second order convergence of the error that is predicted by the theoretical analysis. The effects of approximation of curved boundaries on the computation of eigenvalue problems is also studied in two dimensions. It is shown that the initial mesh used to define the geometry of the physical domain should contain at least $\sqrt{N}$ boundary points to keep the effects of boundary approximation at the level of the error of the finite element approximation, where $N$ is the number of the elements in the final adaptive mesh. Only about $\sqrt[3]{N}$ boundary points in the initial mesh are needed for boundary value problems. This implies that the computation of eigenvalue problems is more sensitive to the boundary approximation than that of boundary value problems.

Numerical resolution of an anisotropic non-linear diffusion problem

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter e, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit e → 0 is demonstrated to be a singular perturbation of the original diffusion equation. To address efficiently this problem, an Asymptotic-Preserving scheme is derived. This numerical method does not require the use of coordinates adapted to the anisotropy direction and exhibits an accuracy as well as a computational cost independent of the anisotropy strength.

Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in R N ( N ≥ 3 )

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem in R N for N≥3 under nonlinear boundary conditions. Utilizing the technique of a differential inequality, lower bounds for the blow-up time are derived when the blow-up does occur under some suitable assumptions.MSC: 35K20, 35K55, 35K65.

Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems

The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal. <br><br> For more information please click the “Full Text” above.

Nonlocal $p$-Laplace equations depending on the $L^p$ norm of the gradient

We are studying a class of nonlinear nonlocal diffusion problems associated with a p-Laplace-type operator, where a nonlocal quantity is present in the diffusion coefficient. We address the issues of existence and uniqueness for the parabolic setting. Then, we study the asymptotic behavior of the solution for large time. For this purpose, we introduce and investigate, in detail, the associated stationary problem. Moreover, since the solutions of the stationary problem are also critical points of some energy functional, we make a classification of its critical points.

Compact Dynamic Modeling for Fast Simulation of Nonlinear Heat Conduction in Ultra-Thin Chip Stacking Technology

A novel nonlinear model order reduction method is proposed for constructing one-port dynamic compact models of nonlinear heat diffusion problems for ultra-thin chip stacking technology. The method leads to models of small state-space dimensions, which allow accurately reconstructing the whole time evolution of the temperature field due to an arbitrary power waveform of practical interest. The approach is also efficient, since the computational time/memory requirements for constructing each dynamic compact model is about one order of magnitude lower than that corresponding to a single 3-D finite element method transient simulation of a nonlinear problem.

Dynamic chaos in the tunnelling ionization produced by a strong low-frequency electromagnetic field

Ionization of atoms by a strong low-frequency linearly polarized electromagnetic field (the photon energy is small compared to the atomic ionization potential) is considered under new conditions compared to the well known Keldysh approach. The field strength is supposed to be small in comparison to the atomic field strength. But the Coulomb interaction of an electron with atomic core is assumed to be of the same order of magnitude as the interaction between an electron and the external electromagnetic field. It was shown that then classical electron motion in the continuum becomes chaotic (this is so-called dynamic chaos). Using the averaging procedure of Chirikov about the chaotic variation of the phase of motion, the considered Newton problem is transformed into the problem of nonlinear electron diffusion over energy scale. In this work we derive the classical electron energy averaged over fast chaotic oscillations of an electron in the final continuum state which takes into account both the Coulomb field and electromagnetic field. This energy is used for analytic calculation of the ionization rate of the ground atomic state into the low lying continuum state based on the Landau–Dykhne approximation (with exponential accuracy). We found that the ionization rate depends significantly on the field frequency. When field frequency decreases, the well known tunnelling limit has been obtained, and then the ionization rate does not depend on the field frequency.