Giovanni Samaey

Equation-Free Multiscale Computation: Algorithms and Applications

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.

A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates

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.

Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics

We present a micro-macro method for the simulation of large elastic deformations of plant tissue. At the microscopic level, we use a mass-spring model to describe the geometrical structure and basic properties of individual plant cells. The macroscopic domain is discretized using standard finite elements, in which the macroscopic material properties (the stress-strain relation) are not given in analytical form, but are computed using the microscopic model in small subdomains, called representative volume elements (RVEs), centered around the macroscopic quadrature points. The boundary conditions for these RVEs are derived from the macroscopic deformation gradient. The computation of the macroscopic stress tensor is based on the definition of virial stress, as defined in molecular dynamics. The anisotropic Eulerian elasticity tensor is estimated using a forward finite difference approximation for the Truesdell rate of the Cauchy stress tensor. We investigate the influence of the size of the RVE and the boundary conditions. This multi-scale method converges to the solution of the full microscopic simulation, for both globally and adaptively refined finite element meshes, and achieves a significant speedup compared to the full microscopic simulation.

Mechanisms of soft cellular tissue bruising. A particle based simulation approach

This paper is concerned with modeling the mechanical behavior of cellular tissue in response to dynamic stimuli. The objective is to investigate the formation of bruises and other damage in tissue under excessive loading. We propose a particle based model to numerically study cells and aggregates of cells described on to subcellular detail. The model focuses on a parenchyma cell type in which two important features are present: the cells interior liquid-like phase inducing hydrodynamic phenomena; and the cell wall, a viscoelastic-plastic solid membrane that encloses the protoplast. The cell fluid is modeled by a Smoothed Particle Hydrodynamics (SPH) technique, while for the cell wall and cell adhesion a nonlinear discrete element model is proposed. Failure in the system is addressed to either cell wall rupture or to debonding of the middle lamella. We show that the model is able to reproduce experimental data of quasistatic compression, and investigate the role of the protoplasm viscosity and the cellular structure on the dynamics of the aggregate system. This indicates that a high viscosity causes better guidance of mechanical stresses through the tissue and can result in a higher penetration of damage, whereas low values will cause more local bruising effects.

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.

Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit

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.

Damping factors for the gap-tooth scheme

An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, the gap-tooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain; it only uses appropriately initialized simulations of the available microscopic model in a number of small boxes. For some model problems, including numerical homogenization, the scheme is essentially equivalent to a finite difference scheme, provided we define appropriate algebraic constraints in each time-step to impose near the boundary of each box. Here, we demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffers that “shield” over relatively short time intervals the dynamics inside each box from boundary effects. We explore and quantify the behavior of these schemes systematically through the numerical computation of damping factors of the corresponding coarse time-stepper, for which no closed formula is available.

A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model over the complete time interval. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely, singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The error bound depends on the modeling error ...

Development of self-organising emergent applications with simulation-based numerical analysis

The goal of engineering self-organising emergent systems is to acquire a macroscopic system behaviour solely from autonomous local activity and interaction. Due to the non-deterministic nature of such systems, it is hard to guarantee that the required macroscopic behaviour is achieved and maintained. Before even considering a self-organising emergent system in an industrial context, e.g. for Automated Guided Vehicle (AGV) transportation systems, such guarantees are needed. An empirical analysis approach is proposed that combines realistic agent-based simulations with existing scientific numerical algorithms for analysing the macroscopic behaviour. The numerical algorithm itself obtains the analysis results on the fly by steering and accelerating the simulation process according to the algorithms goal. The approach is feasible, compared to formal proofs, and leads to more reliable and valuable results, compared to mere observation of simulation results. Also, the approach allows to systematically analyse the macroscopic behaviour to acquire macroscopic guarantees and feedback that can be used by an engineering process to iteratively shape a self-organising emergent solution.

A high-order asymptotic-preserving scheme for kinetic equations using projective integration

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp., parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge--Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyze stability and consistency, and illustrate with numerical results.