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Subject: finite volume methods ×

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    Multiscale coupling of one-dimensional vascular models and elastic tissues
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heltai, Luca; Caiazzo, Alfonso; Müller, Lucas O.
    We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. The pressure and velocity of the blood in the vessels are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.
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    Entropy and convergence analysis for two finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Gaudeul, Benoît; Fuhrmann, Jürgen
    In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.
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    Comparison and numerical treatment of generalised Nernst-Planck models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fuhrmann, Jürgen
    In its most widespread, classical formulation, the Nernst-Planck-Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson-Boltzmann system describing the thermodynamical equilibrium is a stationary state of the discretized time-dependent generalized Nerns-Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach.