A three-dimensional model of a gap junction

Gap junctions are effective electric couplings between neurons and form a very important way of communication between them. Since they can be considered as the points on the neuron’s membrane on which for example dendrites of different cells become one piece, in three dimensions they can be modelled by observing this property in the created geometry. Thus they can be easily made part in an already existing 3-dimensional model for signal propagation on the neuron’s membrane, if the geometries are chosen in such a way that they respect the blending of the membranes. A small network of two cells was created, which blend in their dendrites and a simulation of the three-dimensional model was carried out which reveals the fast transmission of the signal from one cell to the other

Considerations on incompressibility in Linear Elasticity

The classical way to treat incompressible linear elastic materials is to use the inverse constitutive relationship (strain as a function of the stress), based on the compliance tensor, in place of the direct constitutive equation (stress as a function of strain), based on the elasticity (stiffness) tensor. This is because the elasticity tensor is affected by a diverging bulk modulus, requiring in order to allow the material to sustain any hydrostatic load, and is therefore not defined. In this work we show that there is a part of the elasticity tensor that can be saved also for incompressible materials, by filtering the components that deal with hydrostatic loads. The procedure is based on the treatment of incompressibility by means of the constant of isochoric motion, i.e. of conservation of volume, and fourth-order tensor algebra

Bridging the gap between geometric and algebraic multi-grid methods

In this paper, a multi-grid solver for the discretisation of partial differential equations on complicated domains is developed. The algorithm requires as input the given discretisation only instead of a hierarchy of discretisations on coarser grids. Such auxiliary grids and discretisations are generated in a black-box fashion and are employed to define purely algebraic intergrid transfer operators. The geometric interpretation of the algorithm allows one to use the framework of geometric multigrid methods to prove its convergence. The focus of this paper is on the formulation of the algorithm and the demonstration of its efficiency by numerical experiments, while the analysis is carried out for some model problems

Filtering Algebraic Multigrid and Adaptive Strategies

Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically

Numerical simulation of skin transport using Parareal

In silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination’s performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal’s convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed