# Stochastic & Numerical Algorithm Development

Computational techniques are the foundation of all the work done within the Computational Modelling Group. Indeed, many of the physical systems investigated by various members of the group were intractable before the advent of modern computer hardware and efficient numerical algorithms. Besides using the latest computational techniques, many members of the group take an active interest in the mathematical development, formulation and practical implementation of *new* algorithms within the various fields of interest.

As an example, note that the particle dynamics of many processes of interest (for example, in the Nanoparticles and Particle Processes themes), are governed by a population balance. The simplest form of such a population balance is the Smoluchowski coagulation equation. A more detailed - and physically useful - model is formed by adding additional terms to this equation to account for new particles entering the system through formation in the gas phase, surface growth through contact with gaseous species and coalescence through sintering. The governing equation thus modified is a bivariate population balance with volume and surface area as the two internal coordinates. This can be solved to give particle size distributions at various locations in, for example, a flame.

Bivariate population balances are computationally intensive to solve using standard numerical techniques. To overcome this obstacle, the group has developed a Stochastic Particle Algorithm for solving such equations. This represents a breakthrough in numerical efficiency for solving population balances. The efficiency of the algorithm has enabled the simulation of a greater number of internal coordinates: in the extreme case it is now possible to simulate the full spatial structure of the agglomerates. This allows the visualisation of the simulated particles and subsequent direct comparison with TEM micrographs.

## Recent Associated Preprints

192: A high-dimensional, stochastic model for twin-screw granulation. Part 2: Numerical methodology

ref: Technical Report 192, c4e-Preprint Series, Cambridge, 2017 by Andrew D McGuire, Sebastian Mosbach, Kok Foong Lee, Gavin Reynolds, and Markus Kraft

189: An ontology framework for information modeling and management of eco-industrial parks

ref: Technical Report 189, c4e-Preprint Series, Cambridge, 2017 by Li Zhou, Chuan Zhang, Iftekhar A. Karimi, and Markus Kraft

ref: Technical Report 186, c4e-Preprint Series, Cambridge, 2017 by Alastair J. Smith, Clive G. Wells, and Markus Kraft

## Recent Associated Publications

A new iterative scheme for solving the discrete Smoluchowski equation,

Alastair J. Smith, Clive G. Wells, and Markus Kraft, Journal of Computational Physics 352, 373-387, (2018)

Evaluating smart sampling for constructing multidimensional surrogate models,

Sushant S. Garud, Iftekhar A. Karimi, George Brownbridge, and Markus Kraft, Computers and Chemical Engineering 108, 276-288, (2018)

Philipp Buerger, Jethro Akroyd, Sebastian Mosbach, and Markus Kraft, Combustion and Flame 187, 105-121, (2018)

## Recent Associated Presentations

- Presentation by Markus Kraft Download: PDF (963.15 KB)

- Presentation by Markus Kraft
- Invited presentation by Markus Kraft Download: PDF (1.93 MB)

- Invited presentation by Markus Kraft

## Funding

Funding has generously been provided by EPSRC, Toyota, CMCL Innovations and CREATE.