In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form-hence the term equation-free. We selectively discuss basic algorithms and underlying principles and illustrate the approach through representative applications. We also discuss potential difficulties and outline areas for future research.
An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an "equationfree framework" has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with non-linear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.
This paper is concerned with addressing how plant tissue mechanics is related to the micromechanics of cells. To this end, we propose a mesh-free particle method to simulate the mechanics of both individual plant cells (parenchyma) and cell aggregates in response to external stresses. The model considers two important features in the plant cell: (1) the cell protoplasm, the interior liquid phase inducing hydrodynamic phenomena, and (2) the cell wall material, a viscoelastic solid material that contains the protoplasm. In this particle framework, the cell fluid is modeled by smoothed particle hydrodynamics (SPH), a mesh-free method typically used to address problems with gas and fluid dynamics. In the solid phase (cell wall) on the other hand, the particles are connected by pairwise interactions holding them together and preventing the fluid to penetrate the cell wall. The cell wall hydraulic conductivity (permeability) is built in as well through the SPH formulation. Although this model is also meant to be able to deal with dynamic and even violent situations (leading to cell wall rupture or cell-cell debonding), we have concentrated on quasi-static conditions. The results of single-cell compression simulations show that the conclusions found by analytical models and experiments can be reproduced at least qualitatively. Relaxation tests revealed that plant cells have short relaxation times (1 micros-10 micros) compared to mammalian cells. Simulations performed on cell aggregates indicated an influence of the cellular organization to the tissue response, as was also observed in experiments done on tissues with a similar structure.
We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.
This paper is concerned with addressing how biological cells react to mechanical impulse. We propose a particle based model to numerically study the mechanical response of these cells with subcellular detail. The model focuses on a plant cell in which two important features are present: (1) the cell's interior liquidlike phase inducing hydrodynamic phenomena, and (2) the cell wall, a viscoelastic solid membrane that encloses the protoplast. In this particle modeling framework, the cell fluid is modeled by a standard smoothed particle hydrodynamics (SPH) technique. For the viscoelastic solid phase (cell wall), a discrete element method (DEM) is proposed. The cell wall hydraulic conductivity (permeability) is built in through a constitutive relation in the SPH formulation. Simulations show that the SPH-DEM model is in reasonable agreement with compression experiments on an in vitro cell and with analytical models for the basic dynamical modes of a spherical liquid filled shell. We have performed simulations to explore more complex situations such as relaxation and impact, thereby considering two cell types: a stiff plant type and a soft animal-like type. Their particular behavior (force transmission) as a function of protoplasm and cell wall viscosity is discussed. We also show that the mechanics during and after cell failure can be modeled adequately. This methodology has large flexibility and opens possibilities to quantify problems dealing with the response of biological cells to mechanical impulses, e.g., impact, and the prediction of damage on a (sub)cellular scale.
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