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Models for capturing the penetration of a diffusant concentration into rubber: Numerical analysis and simulation
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-6564-3598
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 [en]
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: urn:nbn:se:kau:diva-98719DOI: 10.59217/aetx1744ISBN: 978-91-7867-442-8 (print)ISBN: 978-91-7867-443-5 (electronic)OAI: oai:DiVA.org:kau-98719DiVA, id: diva2:1841805
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
List of papers
1. A Moving Boundary approach of Capturing diffusants Penetration into Rubber: FEM Approximation and Comparison with laboratory Measurements
Open this publication in new window or tab >>A Moving Boundary approach of Capturing diffusants Penetration into Rubber: FEM Approximation and Comparison with laboratory Measurements
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2021 (English)In: KGK Kautschuk, Gummi, Kunststoffe, ISSN 0948-3276, Vol. 74, no 5, p. 61-69Article in journal (Refereed) Published
Abstract [en]

To model the penetration of diffusants into dense and foamed rubbers a moving -boundary scenario is proposed. After a brief discussion of scaling arguments, we present a finite element approximation of the moving boundary problem. To overcome numerical difficulties due to the a priori unknown motion of the diffusants penetration front, we transform the governing model equations from the physical domain with moving unknown boundary to a fixed fictitious domain. We then solve the transformed equations by the finite element method and explore the robustness of our approximations with respect to relevant model parameters. Finally, we discuss numerical estimations of the expected large -time behavior of the material.

Place, publisher, year, edition, pages
Huethig GmbH & Co. KG, 2021
Keywords
Moving boundary problem; Swelling; Finite element method
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-87401 (URN)000711597500011 ()
Available from: 2021-11-25 Created: 2021-11-25 Last updated: 2024-03-01Bibliographically approved
2. Error estimates for semi-discrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber
Open this publication in new window or tab >>Error estimates for semi-discrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber
2022 (English)In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 19, no 1, p. 101-125Article in journal (Refereed) Published
Abstract [en]

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.

Place, publisher, year, edition, pages
ISCI-INST SCIENTIFIC COMPUTING & INFORMATION, 2022
Keywords
Moving boundary problem, finite element method, method of lines, a priori error estimate, a posteriori error estimate, diffusion of chemicals into rubber
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-88250 (URN)000767585800006 ()2-s2.0-85128704007 (Scopus ID)
Funder
Swedish Research Council, 2018-03648Knowledge Foundation, 2019-0213
Available from: 2022-01-28 Created: 2022-01-28 Last updated: 2024-03-01Bibliographically approved
3. Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants
Open this publication in new window or tab >>Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants
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
Keywords
Moving-boundary problem, Finite element approximation, Fully discrete approximation, A priori error estimate
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-92518 (URN)10.1016/j.amc.2022.127733 (DOI)000923199000001 ()2-s2.0-85144575490 (Scopus ID)
Funder
Swedish Research Council, 018-03648Knowledge Foundation, 019-021
Available from: 2022-11-20 Created: 2022-11-20 Last updated: 2024-03-01Bibliographically approved
4. Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers
Open this publication in new window or tab >>Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers
2023 (English)In: Probabilistic Engineering Mechanics, ISSN 0266-8920, E-ISSN 1878-4275, Vol. 74, article id 103546Article in journal (Refereed) Published
Abstract [en]

For certain materials science scenarios arising in rubber technology, one-dimensional moving boundary problems with kinetic boundary conditions are capable of unveiling the large-time behavior of the diffusants penetration front, giving a direct estimate on the service life of the material. Driven by our interest in estimating how a finite number of diffusant molecules penetrate through a dense rubber, we propose a random walk algorithm to approximate numerically both the concentration profile and the location of the sharp penetration front. The proposed scheme decouples the target evolution system in two steps: (i) the ordinary differential equation corresponding to the evaluation of the speed of the moving boundary is solved via an explicit Euler method, and (ii) the associated diffusion problem is solved by a random walk method. To verify the correctness of our random walk algorithm we compare the resulting approximations to computational results based on a suitable finite element approach with a controlled convergence rate. Our numerical results recover well penetration depth measurements of a controlled experiment designed specifically for this setting.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Explicit euler method, Finite element approximation, Moving boundary problem with kinetic condition, Random walk approximation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-97192 (URN)10.1016/j.probengmech.2023.103546 (DOI)001108952000001 ()2-s2.0-85175365511 (Scopus ID)
Funder
Swedish Research Council, VR 2018-03648Knowledge Foundation, KK 2019-0213; KK 2020-0152
Available from: 2023-10-27 Created: 2023-10-27 Last updated: 2024-03-01Bibliographically approved
5. Numerical study of a strongly coupled two-scale system with nonlinear dispersion
Open this publication in new window or tab >>Numerical study of a strongly coupled two-scale system with nonlinear dispersion
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

Thinking of flows crossing through regular porous media, we numerically explore the behavior of weak solutions to a two-scale elliptic-parabolic system that is strongly coupled by means of a suitable nonlinear dispersion term. The two-scale system of interest originates from the fast-drift periodic homogenization of a nonlinear convective-diffusion-reaction problem, where the structure of the non-linearity in the drift fits to the hydrodynamic limit of a totally asymmetric simple exclusion process for a population of particles. In this article, we focus exclusively on numerical simulations that employ two decoupled approximation schemes, viz. ''scheme 1" -- a Picard-type iteration - and ''scheme 2" -- a time discretization decoupling. Additionally, we describe a computational strategy which helps to drastically improve computation times. Finally, we provide several numerical experiments to illustrate what dispersion effects are introduced by a specific choice of microstructure and model ingredients.

Keywords
Nonlinear dispersion, Iterative scheme, FEM approximations, Two-scale systems, Weak solutions, Numerical simulation
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-98724 (URN)
Available from: 2024-03-01 Created: 2024-03-01 Last updated: 2024-03-01

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Nepal, Surendra

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