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Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-6564-3598
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-3156-1420
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-1160-0007
2023 (English)In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 442, article id 127733Article in journal (Refereed) Published
Abstract [en]

We present a fully discrete scheme for the numerical approximation of a moving-boundary problem describing diffusants penetration into rubber. Our scheme utilizes the Galerkin finite element method for the space discretization combined with the backward Euler method for the time discretization. Besides dealing with the existence and uniqueness of solution to the fully discrete problem, we assume sufficient regularity for the solution to the target moving boundary problem and derive a a priori error estimates for the mass concentration of the diffusants, and respectively, for the position of the moving boundary. Our numerical results illustrate the obtained theoretical order of convergence in physical parameter regimes.

Place, publisher, year, edition, pages
Elsevier, 2023. Vol. 442, article id 127733
Keywords [en]
Moving-boundary problem, Finite element approximation, Fully discrete approximation, A priori error estimate
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-92518DOI: 10.1016/j.amc.2022.127733ISI: 000923199000001Scopus ID: 2-s2.0-85144575490OAI: oai:DiVA.org:kau-92518DiVA, id: diva2:1712070
Funder
Swedish Research Council, 018-03648Knowledge Foundation, 019-021Available from: 2022-11-20 Created: 2022-11-20 Last updated: 2024-03-01Bibliographically approved
In thesis
1. Models for capturing the penetration of a diffusant concentration into rubber: Numerical analysis and simulation
Open this publication in new window or tab >>Models for capturing the penetration of a diffusant concentration into rubber: Numerical analysis and simulation
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Understanding the transport of diffusants into rubber plays an important role in forecasting the material's durability. In this regard, we study different models, conduct numerical analysis, and present simulation results that predict the evolution of the penetration front of diffusants.

We start with a moving-boundary approach to model this phenomenon, employing a numerical scheme to approximate the diffusant profile and the position of the moving boundary capturing the penetration front. Our numerical scheme utilizes the Galerkin finite element method for space discretization and the backward Euler method for time discretization. We analyze both semi-discrete and fully discrete approximations of the weak solution to the model equations, proving error estimates and demonstrating good agreement between numerical and theoretical convergence rates. Numerically approximated penetration front of the diffusant recovers well the experimental data.  

As an alternative approach to finite element approximation, we introduce a random walk algorithm that employs a finite number of particles to approximate both the diffusant profile and the location of the penetration front. The transport of diffusants is due to unbiased randomness, while the evolution of the penetration front is based on biased randomness. Simulation results obtained via the random walk approach are comparable with the one based on the finite element method.

In a multi-dimensional scenario, we consider a strongly coupled elliptic-parabolic two-scale system with nonlinear dispersion that describes particle transport in porous media. We construct two numerical schemes approximating the weak solution to the two-scale model equations. We present simulation results obtained with both schemes and compare them based on computational time and approximation errors in suitable norms. By introducing a precomputing strategy, the computational time for both schemes is significantly improved.

Abstract [en]

Understanding the transport of diffusants into rubber plays an important role in forecasting the material's durability. In this regard, we study different models, conduct numerical analysis, and present simulation results that predict the evolution of diffusant penetration fronts. We employ a moving-boundary approach to model this phenomenon, utilizing a numerical scheme based on the Galerkin finite element method combined with the backward time discretization, to approximate the diffusant profile and the position of the penetration front. Both semi-discrete and fully discrete approximations are analyzed, demonstrating good agreement between numerical and theoretical convergence rates. Numerically approximated diffusants penetration front recovers well the experimental data. We introduce a random walk algorithm as an alternative tool to the finite element method, showing comparable results to the finite element approximation. In a multi-dimensional scenario, we consider a strongly coupled elliptic-parabolic two-scale system with nonlinear dispersion, describing the particle transport in a porous medium. We present two numerical schemes and compare them based on computational time and approximation errors. A precomputing strategy significantly improves computational efficiency.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2024. p. 23
Series
Karlstad University Studies, ISSN 1403-8099 ; 2024:8
Keywords
transport of diffusants, moving-boundary problem, finite element method, a priori and a posteriori error estimates, random walk method, two-scale coupled system
National Category
Mathematics Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-98719 (URN)10.59217/aetx1744 (DOI)978-91-7867-442-8 (ISBN)978-91-7867-443-5 (ISBN)
Public defence
2024-04-16, Eva Eriksson lecture hall, 21A342, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2024-03-26 Created: 2024-03-01 Last updated: 2024-03-26Bibliographically approved

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Nepal, SurendraWondmagegne, YosiefMuntean, Adrian

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