Stationary Diffusion Problems Research Articles

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

Study of Hydrogen Embrittlement in Pipelines and Nuclear Power Plants: Estimates for Durability

Due to the increasing need to further develop the world gas and oil industry and the increased public attention to clean energy sources, studying and preventing Hydrogen Induced Cracking is one of the main safety concerns in nuclear power plants, oil pipelines and platforms. In this article, the growth and incubation times for internal Hydrogen Induced Cracks (HIC) are examined. Specifically, these times are modeled in two separate phases - the first phase (I) is a long time approximation, when the crack growth is believed to be slow such that the equilibrium state for gas concentration establishes instantaneously, and the stationary diffusion problem can be solved for each moment of time. The second phase (II) is a short time approximation, when the crack growth is rapid and the concentration of atomic hydrogen is dependent on time. Closed-form solutions are obtained in both cases and are then coupled using a Padé approximation.

Investigation of the evaporation in the boundary layer

THE PURPOSE. Consider a stationary diffusion problem when a pure liquid evaporates from a flat surface of evaporation into a laminar boundary layer of a forced gas flow (in the absence of deepening of the evaporation surface and wave formation on it) at the number. In the classical model of the diffusion problem of the flow of mass from a flat surface into the laminar boundary layer, only the additional slowing down effect arising in this case is taken into account. However, the resulting solution does not correspond to the general case of evaporation, since in this case the mass transfer can significantly depend on the thermal conditions of the problem, conjugate in phases; in criterion form, this circumstance is expressed by the appearance of an additional parameter [1-3]. Note that this parameter is related to the value of the derivative of the relative concentration along the transverse coordinate on the evaporation surface. In the course of the proposed solution, the temperature of the evaporation surface and, accordingly, the value of this parameter were taken constant. METHODS. When solving the problem, we used approximate numerical methods for integrating the diffusion equation (Euler's method, integro-differential equation method, and also the method of successive approximations). In this case, the retarding effect of the vapor flow from the surface of the phase transition was assumed to be relatively insignificant in our case (which corresponded to the experimental data used in [1-3]. RESULTS. The article analyzes the well-known classical solution of the diffusion equation according to the Hartnett - Eckert model and notes that the result obtained in this case does not correspond to the general case of evaporation, when the mass transfer in the gas phase also depends on the complex. Based on the solution obtained in our work, we come to the conclusion that the effect of this parameter manifests itself in an increase in the thickness of the diffusion boundary layer. In addition, this effect is also associated with the value of the longitudinal coordinate, being more noticeable at its small values. CONCLUSION. The indicated evaporation pattern can be physically explained by a relatively larger amount of evaporated substance than in the “standard” case (since values, in turn, are associated with higher values of the evaporation surface temperature). It can also be assumed that in the region of the gas flow immediately adjacent to the evaporation surface, these factors manifest themselves in a similar way in the case of turbulent flows.

Direct and inverse problems for the Poisson equation with equality of flows on a part of the boundary

ABSTRACTIn the paper we consider a stationary diffusion problem described by the Poisson equation. The problem is considered in a model domain, chosen as a half disk. Classical Dirichlet boundary conditions are set on the arc of the circle. New nonlocal boundary conditions are set on the bottom base. The first condition means the equality of flows through opposite radii, and the second condition is the proportionality of distribution densities on these radii with a variable coefficient of proportionality. Uniqueness and existence of the classical solution to the problem are proved. An inverse problem for the solution to the Poisson equation and its right-hand part depending only on an angular variable are considered. As an additional condition we use the boundary overdetermination. Inverse problems to the Dirichlet and Neumann problems, and to problems with nonlocal conditions of the equality of flows through the opposite radii are considered. The well-posedness of the formulated inverse problems is proved.

Error estimates on a finite volume method for diffusion problems with interface on rectangular grids

The finite volume methods are frequently employed in the discretization of diffusion problems with interface. In this paper, we firstly present a vertex-centered MACH-like finite volume method for solving stationary diffusion problems with strong discontinuity and multiple material cells on the Eulerian quadrilateral grids. This method is motivated by Frese [No. AMRC-R-874, Mission Research Corp., Albuquerque, NM, 1987]. Then, the local truncation error and global error estimates of the degenerate five-point MACH-like scheme are derived by introducing some new techniques. Especially under some assumptions, we prove that this scheme can reach the asymptotic optimal error estimate O(h2|ln h|) in the maximum norm. Finally, numerical experiments verify theoretical results.

Scalable two-level preconditioning and deflation based on a piecewise constant subspace for (SIP)DG systems for diffusion problems

This paper is focused on the preconditioned Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations for stationary diffusion problems. In particular, it concerns two-level preconditioning strategies where the coarse space is based on piecewise constant DG basis functions. In this paper, we show that both the two-level preconditioner and the corresponding BNN (or ADEF2) deflation variant yield scalable convergence of the CG method (independent of the mesh element diameter). These theoretical results are illustrated by numerical experiments.

On the dependence of the critical crystallization rate on the initial impurity concentration in melt

The dependence of the critical crystallization rate on the initial impurity concentration in melt is derived by determining the condition at which a nonplanar solution to the stationary diffusion problem arises. It is suggested that the conditions at which defects arise at the interface differ from those obtained when crystallization becomes stationary. The initial transient process of binary melt solidification under crystal pulling at a constant rate has been studied numerically within a 1D model. It is shown that the character of the time dependence of the crystallization rate is determined by the ratio of the crystal pulling rate to solidification rate Vc, when a constitutional supercooling zone is formed in the melt.

Sequential laminates in multiple-state optimal design problems

In the study of optimal design related to stationary diffusion problems with multiple-state equations, the description of the setH={(Aa1,...,Aam):A∈K(θ)}for given vectorsa1,...,am∈ℝd(m<d) is crucial.K(θ)denotes all composite materials (in the sense of homogenisation theory) with given local proportionθof the first material. We prove that the boundary ofHis attained by sequential laminates of rank at mostmwith matrix phaseαIand coreβI(α<β). Examples showing that the information on the rank of optimal laminate cannot be improved, as well as the fact that sequential laminates with matrix phaseαIare preferred to those with matrix phaseβI, are presented. This result can significantly reduce the complexity of optimality conditions, with obvious impact on numerical treatment, as demonstrated in a simple numerical example.

Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems

A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimal-order estimate for the point-wise error and a corresponding a posteriori error bound. The proofs are based on weighted -norm error estimates for discrete Green functions as already known for the ‘continuous’ finite element method.

On effective diffusivity of periodic non‐homogeneous solid media

AbstractDiffusion in a non‐homogeneous medium with periodic spatial variations of diffusivity and driving force is analyzed in detail. After a time t ≪ τ (τ being time to attain local quasi‐stationarity) diffusion proceeds like in a homogeneous medium with an effective diffusivity D̃ij. Correct procedure of D̃ij calculation includes solution of a stationary diffusion problem in a periodic unit cell with special boundary conditions. Methods of exact calculation and estimation of D̃ij are discussed, some simple estimates are presented. Possible applications of the theory are sketched.

Influence du gravage des joints de grains sur la vitesse de fluage diffusion des fils à structure bambou et la détermination corrélative de la tension superficielle

Abstract Influence of grain bouwdary grooving on the diffusional creep rate of bamboo-structure wires and the correlated determination of surface tension By solving the stationary diffusion problem with boundary conditions which account for grain boundary grooving and the effect of surface and grain boundary tensions on the local chemical potential, the equation of the diffusional creep rate of bamboo-structure wires is derived in the cases where diffusion occurs either through the volume or along the grain boundaries. The calculation shows that, under usual experimental conditions, the grain boundary grooving does not affect the structure of the classical diffusion-controlled creep-rate equation, but significantly alters the correlated determination of surface tension. The corrections which must be made to the previous determinations of surface tension obtained from the zero creep-rate stress of such wires without accounting for grooving, are also calculated as a function of the groove width and of the grain length. Under usual experimental conditions they are of the order of +50% to +60%, but they can reach + 100% when the grooves are rather shallow. Finally the diffusion creep of thin sheets and sintering of particles and wires is discussed. En résolvant le problème de diffusion des lacunes en régime stationnaire avec des conditions aux limites qui tiennent compte du gravage des joints de grains et de l'influence des tensions superficielle et intergranulaire sup le potentiel chimique local, on prévoit l'expression de la vitesse de fluage diffusion des fils à structure bambou dans les cas o[ugrave] la diffusion s'effectue uniquement en volume ou le long des joints. Ce faisant, nous montrons que dans les conditions expérimentales courantes le gravage des joints n'affecte pas la structure de l'expression classique de la vitesse de fluage contrôlée par la diffusion, mais altère de façon significative la détermination corrélative de la tension superficielle. On calcole en fonction de la largeur des sillons de gravage et de la longueur des grains les corrections qu'il convient d'apporter aux détterminations antétrieures de la tension superficielle obtenues sans tenir compte du gravage. Celles-ci sont de l'ordre de + 50% à + 60%, dans les conditions leg plus habituelles, mais peuvent atteindre + 100% quand les gravages sont peu profonds. Puis on étend la cliscussion au fluage diffusion des feuilles et au frittage des poudres et cles fils.