Ward Melis defended his PhD on Projective integration for hyperbolic conservation laws and multiscale kinetic equations on March 23, 2017. More information, including the thesis, can be found here.
Annelies Lejon defended her PhD on Multiscale simulation of tumor growth: projective integration and variance reduction on December 5, 2016. More information, including the thesis, can be found here.
We are involved in the project InterPoCo (Interfaces in polymer composites), which is a collaboration between the five Flemish universities and Flamac. At KU Leuven, this involves collaboration with Dirk Roose (NUMA, Dept. Computer Science) and Erik Nikes (Dept. of Chemistry). Our role in the project is to:
- Create polymer network structures, starting from an ensemble of monomers and a binding agent.
- Perform virtual experiments on the mechanical properties of these networks (such as stress-strain curves or glass transition temperatures).
In the network creation phase, we used iterative Boltzmann inversion to obtain coarse-grained force fields, and we made a number of algorithmic contributions:
- Performing the polymerisation reaction at the coarse-grained level.
- Backmapping the coarse-grained polymer structure to the atomistic level.
The work builds heavily on the Adaptive Resolution scheme AdResS.
- J. Krajniak, S. Pandiyan, E. Nies and G. Samaey, A generic adaptive resolution method for reverse mapping of polymers from coarse-grained to atomistic descriptions, Journal of Chemical Theory and Computation 12:5549–5562, 2016.
- S. Pandiyan, J. Krajniak, G. Samaey, D. Roose and E. Nies, A molecular dynamics study of water transport inside an epoxy polymer matrix, Computational Materials Science 106:29-37, 2015.
Together with Tine Baelmans from the Thermal and Fluids Engineering group, we are working on new numerical methods for the simulation of the plasma edge in next-step nuclear fusion devices.
Two types of particles are modeled in plasma edge codes: the plasma, consisting of charged particles (ions and electrons), and the neutral particles. The plasma can usually be described with a Navier-Stokes-like fluid model, discretized in space with a suitable finite volume (FV) method. For the neutrals, however, a more microscopic, kinetic description is necessary, in which the particle distribution is modeled in a position-velocity phase space. This is because the neutral gas is so rarefied that the fluid approach does not hold for this component. Further, the velocity distribution of the neutral particles is typically so far from equilibrium that a fluid description, based on density, momentum and energy, is not sufficiently accurate. Due to the additional dimensions in velocity space, the kinetic neutral model is solved via Monte Carlo (MC) simulation.
Since the plasma and neutral particles interact with each other, the corresponding models need to be coupled, leading to serious computational bottlenecks. Using present-day stopping criteria, simulations for DEMO last approximately 1 year. We are developing and analyzing new computational methods that alleviate the following main problems:
- The statistical noise introduced by the Monte Carlo simulation on the computed neutral density, and – via the coupling – also on the computed plasma solution.
- The increasing importance of simulating in a so-called ‘detached
regime’, in which there is increased interaction of the neutrals with the ions, leading to excessive computation times if each individual interaction needs to be resolved.
- High variance on the simulations because neutral particles that are launched from the target into the interior of the fusion reactor don’t penetrate very far into the plasma. This leads to a high variability of the computed source terms in the interior of the reactor.
- N. Horsten, G. Samaey, and M. Baelmans, Development and assessment of 2D fluid neutral models that include atomic databases and a microscopic reflection model, 2017. Submitted.
- M. Baelmans, K. Ghoos, P. Börner and G. Samaey, Efficient code simulation strategies for B2-EIRENE, Nuclear Materials and Energy, 2016. In press.
- N. Horsten, W. Dekeyser, G. Samaey and M. Baelmans, Assessment of fluid neutral models for a detached ITER case, Nuclear Materials and Energy, 2016. In press.
- K. Ghoos, W. Dekeyser, G. Samaey and M. Baelmans, Accuracy and convergence of coupled finite-volume/Monte-Carlo codes for plasma edge simulations of nuclear fusion reactors, Journal of Computational Physics 322:162–182, 2016.
- N. Horsten, W. Dekeyser, G. Samaey, P. Börner and M. Baelmans, Fluid neutral model for use in hybrid simulations of a detached case, Contributions to Plasma Physics 56:610-615, 2016.
- N. Horsten, W. Dekeyser, G. Samaey and M. Baelmans, Comparison of fluid neutral models for 1D plasma edge modelling with a finite volume solution of the Boltzmann equation, Physics of Plasmas 23:012510, 2016.
- K. Ghoos, W. Dekeyser, G. Samaey, P. Börner, D. Reiter and M. Baelmans, Accuracy and convergence of coupled finite-volume/Monte-Carlo codes for plasma edge simulations, Contributions to Plasma Physics, 2016. In press.
Kinetic equations describe a particle system in a position-velocity phase space. In the limit of small mean free path (whether in a hydrodynamic or a diffusive scaling), one obtains a macroscopic limiting equation of hyperbolic of diffusive type. Improving this approximation by taking into account kinetic effects often results in a very stiff partial differential equation to solve.
Asymptotic-preserving methods aim at solving such equations by constructing schemes that fall onto a scheme for the limiting macroscopic equation in the appropriate limit, resulting in a discretization of which the computational costs is essentially independent of the mean free path. Projective integration is such a technique that is general, easy to implement, fully explicit and of arbitrary order.
- W. Melis and G. Samaey, Telescopic projective integration for kinetic equations with multiple relaxation times, 2016. Submitted.
- P. Lafitte, W. Melis and G. Samaey, A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws, Journal of Computational Physics 340:1-25, 2017.
- P. Lafitte, A. Lejon and G. Samaey, A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM Journal on Numerical Analysis 54:1-33, 2015.
- P. Lafitte and G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing 34:A579-A602, 2012.
When designing concurrent coupling techniques in multiscale simulation, one often needs to obtain information at the microscopic (detailed) level of description, when the only available information is given at a macroscopic level of description (where only some averaged quantities are modeled). Examples are:
- the probability distributions generated by stochastic differential equations, whose evolution might be approximated at a macroscopic level of description by the evolution of some moments of this distribution (such as mean and variance) – this happens, for instance, in polymeric fluid flow);
- high-dimensional stochastic differential equations with slow and fast degrees of freedom, for which an approximate evolution can be conceived in terms of only the slow degrees of freedom;
- kinetic equations, in which a distribution function in position-velocity phase space may be approximated by some low-order moments over velocity space – this happens, for instance, in biological problems, nuclear fusion or rarified gas flow.
In all these situations (and more), missing information needs to be added at the microscopic scale. We develop and analyze algorithms to do this, based on either a quasi-equilibrium approximation or via minimization of the perturbation to a reference (prior) microscopic state.
- K. Debrabant, G. Samaey and P. Zielinski, A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations, SIAM Journal on Numerical Analysis, 2017. In press.
- F. Legoll, T. Lelièvre, and G. Samaey, A micro/macro parareal algorithm for singularly perturbed ordinary differential equations, SIAM Journal on Scientific Computing 35:A1951-A1986, 2013.
- G. Mazzi, Y. De Decker and G. Samaey, Towards an efficient multiscale modeling of low-dimensional reactive systems: study of numerical closure procedures, Journal of Chemical Physics 137:204115, 2012.
- G. Samaey, T. Lelièvre and V. Legat, A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells, Computers and Fluids, 43:119-133, 2011.
Multiscale models are often used in situations where one is interested in increasing modeling detail to include the effects of phenomena that are microscopic, compared to the macroscopic scale of interest. This usually increased the dimensionality of the model to such an extent that a deterministic simulation becomes intractable and one needs to resort to stochastic (Monte Carlo) simulation methods. Then, a new challenge arises: how can we ensure that the extra accuracy that results from adding the microscopic level of modeling detail is not completely lost in the statistical noise that is inherent in a stochastic Monte Carlo simulation?
In our group, we are working on variance reduction techniques based on control variates in this setting. We either have an approximate macroscopic model available, with which we can perform coupled stochastic simulations, or we couple stochastic simulations with the same microscopic model but related initial conditions.
- A. Lejon, B. Mortier and G. Samaey, Variance-reduced simulation of stochastic agent-based models for tumor growth, 2015. Submitted.
- D. Avitabile, R. Hoyle and G. Samaey, Noise reduction in coarse bifurcation analysis of stochastic agent-based models: an example of consumer lock-in, SIAM Journal on Applied Dynamical Systems 13(4):1583–1619, 2014.
- M. Rousset and G. Samaey, Individual-based models for bacterial chemotaxis in the diffusion asymptotics, Mathematical Models and Methods in Applied Sciences 23:2005-2037, 2013.
- M. Rousset and G. Samaey, Simulating individual-based models for bacterial chemotaxis with asymptotic variance reduction, Mathematical Models and Methods in Applied Sciences 23:2155-2191, 2013.