scholarly journals Algorithms for Numerical Solution to the Problem of Diffusion in Nanoporous Media

2020 ◽  
pp. 44-52
Author(s):  
N.A. Vareniuk ◽  
N.I. Tukalevska
Keyword(s):  
Mathematical Modeling ◽  
Finite Element Method ◽  
Mass Transfer ◽  
Finite Element ◽  
Numerical Solution ◽  
Heterogeneous Media ◽  
Mutual Influence ◽  
Single Substance ◽  
Nonstationary Diffusion ◽  
Element Method

Introduction. Mathematical modeling of mass transfer in heterogeneous media of microporous structure and construction of solutions to the corresponding problems of mass transfer was considered by many authors [1–9, etc.]. In [6, 7] authors proposed a methodology for modeling mass transfer systems and parameter identification in nanoporous particle media (diffusion, adsorption, competitive diffusion of gases, filtration consolidation), which are described by non-classical boundary and initial-boundary value problems taking into account the mutual influence of micro- and macro-transfer flows, heteroporosity, the structure of microporous particles, multicomponent and other factors. In [8, 9] for a mathematical model of nonstationary diffusion of a single substance in a nanoporous medium described in [2] in the form of a multi-scale differential mathematical problem, the classical problems in the weak formulation were obtained. In this paper, algorithms for solving the above mathematical problems are constructed by using the finite element method. The results of the numerical solution of the test problem are presented. The results confirm the efficiency of the developed algorithms. The purpose is to solve a problem of nonstationary diffusion of single substance in nanoporous medium by constructing discretization algorithms using FEM quadratic basis functions. Results. Algorithms for the numerical solution of the problem of nonstationary diffusion of single substance in a nanoporous medium are proposed. Peculiarities of discretization of the region and construction of the matrix of masses, stiffness, and vector of right-hand sides when solving the problem by using FEM are described. The efficiency of the developed algorithms is confirmed by the results of solving a model example. Keywords: mathematical modeling, numerical methods, nonstationary diffusion, nanoporous medium, finite element method.

Physical statement of the problem for a numerical model of soil freezing and heaving taking into account heat and mass transfer

2021 ◽  
pp. 244-256
Author(s):  
Виктор Григорьевич Чеверев ◽  
Евгений Викторович Сафронов ◽  
Алексей Александрович Коротков ◽  
Александр Сергеевич Чернятин
Keyword(s):  
Finite Element Method ◽  
Mass Transfer ◽  
Finite Element ◽  
Numerical Model ◽  
Boundary Conditions ◽  
Heat And Mass Transfer ◽  
Soil Freezing ◽  
The Finite Element Method ◽  
Freezing Front ◽  
Element Method

Существуют два основных подхода решения задачи тепломассопереноса при численном моделировании промерзания грунтов: 1) решение методом конечных разностей с учетом граничных условий (границей, например, является фронт промерзания); 2) решение методом конечных элементов без учета границ модели. Оба подхода имеют существенные недостатки, что оставляет проблему решения задачи для численной модели промерзания грунтов острой и актуальной. В данной работе представлена физическая постановка промерзания, которая позволяет создать численную модель, базирующуюся на решении методом конечных элементов, но при этом отражающую ход фронта промерзания - то есть модель, в которой объединены оба подхода к решению задачи промерзания грунтов. Для подтверждения корректности модели был проделан ряд экспериментов по физическому моделированию промерзания модельного грунта и выполнен сравнительный анализ полученных экспериментальных данных и результатов расчетов на базе представленной численной модели с такими же граничными условиями, как в экспериментах. There are two basic approaches to solving the problem of heat and mass transfer in the numerical modeling of soil freezing: 1) using the finite difference method taking into account boundary conditions (the boundary, for example, is the freezing front); 2) using the finite element method without consideration of model boundaries. Both approaches have significant drawbacks, which leaves the issue of solving the problem for the numerical model of soil freezing acute and up-to-date. This article provides the physical setting of freezing that allows us to create a numerical model based on the solution by the finite element method, but at the same time reflecting the route of the freezing front, i.e. the model that combines both approaches to solving the problem of soil freezing. In order to confirm the correctness of the model, a number of experiments on physical modeling of model soil freezing have been performed, and a comparative analysis of the experimental data obtained and the calculation results based on the provided numerical model with the same boundary conditions as in the experiments was performed.

scholarly journals APPLICATION OF GALERKIN FINITE ELEMENT METHOD IN THE SOLUTION OF 3D DIFFUSION IN SOLIDS

2009 ◽  
Vol 8 (2) ◽  
pp. 79 ◽  
Author(s):  
E. C. Romão ◽  
M. D. De Campos ◽  
J. A. Martins ◽  
L. F. M. De Moura
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Heat Diffusion ◽  
Average Error ◽  
Galerkin Finite Element Method ◽  
Diffusion In Solids ◽  
Galerkin Finite Element ◽  
L2 Norm ◽  
Element Method

This paper presents the numerical solution by the Galerkin Finite Element Method, on the three-dimensional Laplace and Helmholtz equations, which represent the heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used in comparison with the numerical solution. The results analysis was made based on the the L2 Norm (average error throughout the domain) and L¥ Norm (maximum error in the entire domain). The two application results, one of the Laplace equation and the Helmholtz equation, are presented and discussed in order to to test the efficiency of the method.

Comprehensive Mathematical Modeling and Dynamic Simulation of Fixed Bed Reactor with Finite Element Method

2011 ◽  
Author(s):  
Davood Dehestani ◽  
Hung Nguyen ◽  
Fahimeh Eftekhari ◽  
Jafar Madadnia ◽  
Steven Su ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Simulation ◽  
Fixed Bed ◽  
Fixed Bed Reactor ◽  
Element Method
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