| 1. |
- Nepal, Surendra, 1990-
(författare)
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A moving boundary problem for capturing the penetration of diffusant concentration into rubbers : Modeling, simulation and analysis
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We propose a moving-boundary scenario to model the penetration of diffusants into rubbers. Immobilizing the moving boundary by using the well-known Landau transformation transforms the original governing equations into new equations posed in a fixed domain. We solve the transformed equations by the finite element method and investigate the parameter space by exploring the eventual effects of the choice of parameters on the overall diffusants penetration process. Numerical simulation results show that the computed penetration depths of the diffusant concentration are within the range of experimental measurements. We discuss numerical estimations of the expected large-time behavior of the penetration fronts. To have trust in the obtained simulation results, we perform the numerical analysis for our setting. Initially, we study semi-discrete finite element approximations of the corresponding weak solutions. We prove 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. Finally, we present a fully discrete scheme for the numerical approximation of model equations. Our scheme is based on the Galerkin finite element method for the space discretization combined with the backward Euler method for time discretization. In addition to proving the existence and uniqueness of a solution to the fully discrete problem, we also derive a priori error estimates for the mass concentration of the diffusants, and respectively, for the position of the moving boundary that fit to our implementation in Python. Our numerical illustrations verify the obtained theoretical order of convergence in physical parameter regimes.
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| 2. |
- Raveendran, Vishnu
(författare)
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Homogenization of reaction-diffusion problems with nonlinear drift in thin structures
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study the question of periodic homogenization of a variably scaled reaction-diffusion equation with non-linear drift of polynomial type. The non-linear drift was derived as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. We consider three different geometries: (i) Bounded domain crossed by a finitely thin flat composite layer; (ii) Bounded domain crossed by an infinitely thin flat composite layer; (iii) Unbounded composite domain.\end{itemize} For the thin layer cases, we consider our reaction-diffusion problem endowed with slow or moderate drift. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive homogenized evolution equations and the corresponding effective model parameters. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interfaces. As a special scaling, the problem with large drift is treated separately for an unbounded composite domain. Because of the imposed large drift, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. To deal with this special case, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder's fixed point Theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in the unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials which are resistant to slow, moderate, and high velocity impacts.
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| 3. |
- Nepal, Surendra
(författare)
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Models for capturing the penetration of a diffusant concentration into rubber : Numerical analysis and simulation
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Raveendran, Vishnu
(författare)
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Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study the periodic homogenization of reaction-diffusion problems with nonlinear drift describing the transport of interacting particles in composite materials. The microscopic model is derived as the hydrodynamic limit of a totally asymmetric simple exclusion process for a population of interacting particles crossing a domain with obstacles. We are particularly interested in exploring how the scalings of the drift affect the structure of the upscaled model.We first look into a situation when the interacting particles cross a thin layer that has a periodic microstructure. To understand the effective transmission condition, we perform homogenization together with the dimension reduction of the aforementioned reaction-diffusion-drift problem with variable scalings.One particular physically interesting scaling that we look at separately is when the drift is very large compared to both the diffusion and reaction rate. In this case, we consider the overall process taking place in an unbounded porous media. Since we have the presence of a large nonlinear drift in the microscopic problem, we first upscale the model using the formal asymptotic expansions with drift. Then, with the help of two-scale convergence with drift, we rigorously derive the homogenization limit for a similar microscopic problem with a nonlinear Robin-type boundary condition. Additionally, we show the strong convergence of the corrector function. In the large drift case, the resulting upscaled equation is a nonlinear reaction-dispersion equation that is strongly coupled with a system of nonlinear elliptic cell problems. We study the solvability of a similar strongly coupled two-scale system with nonlinear dispersion by constructing an iterative scheme. Finally, we illustrate the behavior of the solution using the iterative scheme.
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