Here is a list of projects that applicants for the MPhil in Scientific Computing can choose from.

**Dr Philip Blakely, Laboratory for Scientific Computing, Department of Physics
Email: pmb39@cam.ac.uk**

Recent developments have resulted in the simultaneous simulation of four states of matter in a single simulation. This uses a wide range of numerical algorithms, from hyperbolic solvers to elliptic solvers, and includes Adaptive Mesh Refinement. All of these algorithms have their own accuracy, stability, and performance characteristics, and ensuring that the best combination of all of them is used is not trivial. This project will involve characterising a multiphysics simulation in terms of its component algorithms, and determining the most efficient solvers for each stage of the process, given a particular required accuracy.

This project could either be a 6-month MPhil research project based on existing software-frameworks, or a longer Ph.D. project with a full development of the student's own framework and algorithmic structure.

**Dr Philip Blakely, Laboratory for Scientific Computing, Department of Physics
Email: pmb39@cam.ac.uk**

Adaptive Mesh Refinement (AMR) is a well-known approach for speeding up Finite-Volume calculations by focusing computational effort on regions where it is most needed. Some large codes exist where GPUs (and similar accelerators) also use AMR. However, the various pay-offs between patch-size, levels of AMR, and load-balancing are different when GPUs are used, due to the different bottlenecks of host-device transfer, memory bandwidth, etc. Research into generally suitable parameters for GPU-accelerated AMR is needed so that we can make good use of the hardware that exists.

This project would be well-suited to a 6-month research project as part of the MPhil. in Scientific Computing.

**Dr Philip Blakely, Laboratory for Scientific Computing, Department of Physics
Email: pmb39@cam.ac.uk**

Much work has been put into accelerating CFD finite-volume HRSC codes by using GPUs and their high memory bandwidth. However, this has generally been done for small systems of equations. For larger systems, particularly multiphase/multiphysics models, the issue of register pressure and limited fast memory becomes important. Research needs to be put into finding ways to optimize for solving these systems on GPUs and will require significant understanding of the NVIDIA hardware and perhaps the PTX assembler language. Investigation of appropriate (and approximate) Riemann solvers, regarding their accuracy and performance on GPUs may also be useful.

This project would be well-suited to a 6-month research project as part of the MPhil. in Scientific Computing.

**Dr Philip Blakely, Laboratory for Scientific Computing, Department of Physics
Email: pmb39@cam.ac.uk**

There is a variety of approaches to modelling discrete interfaces in a finite-volume framework. However, some implementations of the Ghost-Fluid Method require significant work in making them robust for all possible interface types, particularly very thin ones. There are also various algorithms for extrapolating interface ghost-data and it will be useful to investigate the differences in accuracy between them as well as their performance in different scenarios.

This project would be well-suited as either an investigative 6-month MPhil project, or a longer Ph.D. project where new and more robust algorithms could be developed.

**Dr Philip Blakely, Laboratory for Scientific Computing, Department of Physics
Email: pmb39@cam.ac.uk**

Modern multiphysics codes can be applied to a very wide range of problems, and validation is usually performed for a particular application. However, the overall issue of robustness, reproducibility, and regression testing is sometimes overlooked and, for an extremely versatile code, a complete set of tests requires a very long time to evaluate. It would be useful to identify a suitable set of multiphysics tests to which such a code can be applied, that can be run in a fairly short period of time, and identify any code regressions or robustness issues. This will require work on understanding a wide range of multiphysics applications, identification of what simulation features are relevant for reproducibility, and investigation of what code errors are most likely to occur, and devising tests to pick these up.

This project would be well-suited to a 6-month research project as part of the MPhil. in Scientific Computing, and could be expanded into a broader scope for a Ph.D. research project.

**Professor E F Toro, Professor Emeritus of Mathematics, University of Trento, Italy
Visitor, Laboratory for Scientific Computing, Department of Physics **https://www.lsc.phy.cam.ac.uk/

email: eleuterio.toro@unitn.it

The idea is to approximate parabolic equations as hyperbolic equations with stiff source terms, as suggested in [1] for the linear advection-diffusion-reaction equation. More applications of the methodology are found in [2] and [3]. The aim of the project is to apply the methodology to the time-dependent incompressible Navier-Stokes equations, analyse the equations and devise numerical methods to solve them.

**REFERENCES**

[1] Eleuterio F. Toro and Gino I. Montecinos. *Advection-diffusion-reaction equations: hyperbolisation and high-order ADER discretizations.* SIAM Journal of Scientific Computing. Vol. 36, No. 5, pp: A2423-A2457, 2014. https://doi.org/10.1137/130937469

[2] Gino I. Montecinos and Eleuterio F. Toro. *Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes.* Journal of Computational Physics. Vol. 275, 15 October, pp: 415-442, 2014. https://doi.org/10.1016/j.jcp.2014.06.018

[3] Gino I. Montecinos, Lucas O. Müller and Eleuterio F. Toro. *Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes.* Journal of Computational Physics. Vol. 266, pp: 101123, 1 June, 2014. https://doi.org/10.1016/j.jcp.2014.02.013

**Professor E F Toro, Professor Emeritus of Mathematics, University of Trento, Italy****Visitor, Laboratory for Scientific Computing, Department of Physics **https://www.lsc.phy.cam.ac.uk/

email: eleuterio.toro@unitn.it

The aim is to devise flux splitting algorithms of the TV type [1] to solve hyperbolic systems with stiff source terms deriving from relaxation approximations of parabolic equations [2]. Potential examples include the linear advection-diffusion equation and the isothermal compressible Navier-Stokes equations. Second-order extension of the more successful algorithms are envisaged.

**REFERENCES**

[1] Eleuterio F. Toro and M. E. Vázquez-Cendón. *Flux splitting schemes for the Euler equations.* Computers and Fluids. Vol. 70, pp: 1-12, 2012. https://doi.org/10.1016/j.compfluid.2012.08.023

[2] Eleuterio F. Toro and Gino I. Montecinos. *Advection-diffusion-reaction equations: hyperbolisation and high-order ADER discretizations.* SIAM Journal of Scientific Computing. Vol. 36, No. 5, pp: A2423-A2457, 2014. https://doi.org/10.1137/130937469

**Professor E F Toro, Professor Emeritus of Mathematics, University of Trento, Italy
Visitor, Laboratory for Scientific Computing, Department of Physics **https://www.lsc.phy.cam.ac.uk/

email: eleuterio.toro@unitn.it

Most high-order numerical methods for hyperbolic equations, semi-discrete and fully discrete, rely on a monotone (monotone for the scalar case) numerical flux of first-order of accuracy as the building block, e.g. The Godunov first-order flux. In [1] it was suggested to use a TVD (Total variation Diminishing) flux, instead of a first-order monotone flux, ass the building block. Results in [1] and [2] for one-dimensional systems are very encouraging, errors are dramatically reduced. The aim of the project is to extend the use of TVD fluxes, upwind and centred, to the two-dimensional Euler equations on Cartesian meshes and thoroughly assess its advantages over more conventional approaches.

[1] Eleuterio F. Toro and V. A. Titarev. *TVD fluxes for the high-order ADER schemes. *Journal of Scientific Computing. Vol. 24, pp: 285-309, 2005. https://doi.org/10.1007/s10915-004-4790-8

[2] V A Titarev and Eleuterio F. Toro. *ENO and WENO schemes based on upwind and centred TVD fluxes.* Computers and Fluids. Vol. 3, pp: 705-720, 2005. https://doi.org/10.1016/j.compfluid.2004.05.009

**Dr Hrvoje Jasak, Centre for Scientific Computing
Email: hj348@cam.ac.uk**

Inclusion of renewable energy sources into a national power network brings challenges of balancing instantaneous production and consumption of electrical energy. Traditionally, this is resolved by managing the supply to match the demand, but in the presence of substantial renewable supply which should be harvested when available, this is no longer practical.

The most efficient way of achieving the balance is addition of substantial energy storage to the power network. Such examples, based on lithium-ion batteries already exist, but are limited due to fundamental fact that Li-ion cell performance depends on molecular-level properties, such as diffusivity. Alternative battery technology may be more suitable for high power and high capacity applications such as this: for example, liquid metal batteries.

Management of liquid metal batteries in use is a demanding multi-physics problem. It involves multi-phase free surface flow, electromagnetic effects and problems of thermal management. This project is concerned with the construction, efficiency, operating regimes and optimal packaging of large-scale liquid metal batteries in applications related to fixed grid storage

**Dr Hrvoje Jasak, Centre for Scientific Computing
Email: hj348@cam.ac.uk**

In flexibility and automation, additive manufacturing techniques are clearly the future. While 3-D printing is on the way of becoming commonplace, industrial processes requiring high surface and volumetric quality, material strength and absence of defects still hinders wider application of such methods.

The challenge of additive manufacturing is in the fact that most imperfections in final products stems from micro-structural problems, which are in turn caused by insufficiently precise control of heat and mass transfer during phase change (solidification). The heat/mass transfer problem needs to be resolved at the scale of the product itself, while the management of micro-structural properties occurs at significantly smaller scales. The problem can be viewed as the ultimate challenge in fluid-solid interaction, not only involving multiple continua with phase change, but also coupling to crystalline-level effects that in fact govern the quality of the final product.

In this project, macro-scale simulation tools for use in simulation of additive manufacturing processes shall be developed. It involves management of solids, liquids or powders during manufacturing, simulation of phase change with control of liquid melt pools, thermal and residual stresses in the material and two-way coupling with micro-structural models needed to understand the crystalline structure arising from the specific solidification conditions. Such simulation tools can be used to optimise the additive manufacturing and mitigate effects arising from interaction between scales.

**Dr Hrvoje Jasak, Centre for Scientific Computing
Email: hj348@cam.ac.uk**

Numerical simulation is the enabling part in developments and product design in the 21st century. A combination of stringent product performance, life-time and environmental impact of engineering machinery will require real-time simulations and extensive use of digital twins, exploring the performance of real-world systems. This can only be achieved with the use of powerful computer technology.

Projected performance improvement in numerical simulation can no longer simply rely upon ever-increasing compute power. In fact, it is envisaged that future improvement in the power of simulation will come not from faster computers, but mainly from improvements in numerical algorithms adapted to High Performance Computing (HPC) and new hardware paradigms. A good example of this is the problem of implicit linear algebra on exa-scale platforms.

In this project, alternatives in formulation and implementation for high-end linear equation systems and related linear solver, eigen-value decomposition and other algorithms shall be developed for use with exa-scale-capable compute platforms. Improvement in performance shall be achieved by examining alternative numerics algorithms, data layout and parallelisation to achieve order-of-magnitude improvements in this, the most compute-intensive part of most numerical simulators.

**Dr Hrvoje Jasak, Centre for Scientific Computing
Email: hj348@cam.ac.uk**

To date, a bulk of simulations in fluid flow, heat transfer, electromagnetics and associated continuum mechanics phenomena rely on second-order accurate discretisation methods. In most cases, excellent balance of algorithmic simplicity and ease of manipulation created a set of robust simulation tools with known capabilities. Dynamic mesh techniques, topological changes, polyhedral mesh support, adaptive mesh refinement and High Performance Computing (HPC) support is taken for granted, making the existing tools hard to beat.

Having reached the level of available computing power which allows "sufficient mesh resolution" to be applied, a question of how to use this power in the most efficient way arises. Is it better to continue increasing the number of computational points in a low-order method such as second-order Finite Volume Method (FVM), or would a more compute-demanding higher-order method produce a better result?

Having in mind the practical simulation requirements handled with second-order FVM, a new formulation of adaptive high-order Discontinuous Galerkin (DG) Method shall be developed and implemented. It is expected that a combination of high-order temporal and spatial accuracy, flexibility in meshing and HPC support will balance a substantial increase in computational cost per element introduced by such methods.

In this project, a novel DG discretisation with unstructured mesh support and fundamental HPC support will be developed and tested for elliptic and hyperbolic governing laws.

**Dr Hrvoje Jasak, Centre for Scientific Computing
Email: hj348@cam.ac.uk**

Simulation of free surface multi-phase flows is reaching the level of maturity of aerodynamics and turbomachinery simulations, providing robust and accurate numerical simulation tools in naval architecture and ocean engineering. Free surface flows appear at a range of scales, from ship-scale naval hydrodynamics, human-scale wave impact problems to micro-scale laboratory-on-chip flow management. Each of these requires extensions and modifications to the basic model of two incompressible fluids separated by a sharp surface.

In this project, various extensions to the basic free surface flow model and their applications in a multi-physics framework shall be explored. These include n-phase systems, surface tension and related effects, phase compressibility in fluid-solid interaction, phase change (eg. cavitation), multi-phase chemical reaction systems and others.

**Dr Nandan Gokhale, Laboratory for Scientific Computing, Department of Physics
Email: nbg22@cam.ac.uk**

The ability to perform accurate simulations of hypersonic flows around complex geometries is of great practical and scientific interest, for example, in the design of re-entry spacecraft and radio-blackout mitigation systems. Simulating such flows accurately, however, comes with a number of challenges on the mesh generation, physical modelling, and numerical solver fronts. As part of this project, the student will be trained to use and further develop highly parallelisable, cutting-edge, modern computational techniques in conjunction with Adaptive Mesh Refinement and state-of-the art physical modelling to allow the numerical simulation of these flows.

**Dr Nandan Gokhale, Laboratory for Scientific Computing, Department of Physics
Email: nbg22@cam.ac.uk**

Selective Laser Melting (SLM) is an Additive Manufacturing technique that makes use of a high-energy laser to melt and fuse metal powders together. Modern Adaptive Mesh Refinement (AMR) techniques that allow the accurate computational resolution of the small area near the laser front are particularly useful for the simulation of such processes. In this project, the student will be trained in the use of AMR in conjunction with cutting-edge, highly parallelisable, computational algorithms to perform SLM simulations of Titanium alloys that are of particular significance to the Aerospace industry. The project will involve performing both single and multi-layer simulations, and the modelling will consider all the effects relevant to the problem, including the moving energy source, phase change, and radiative and evaporative surface cooling, to name a few.

**Dr Stephen Millmore, Laboratory for Scientific Computing, Department of Physics
Email: stm31@cam.ac.uk**

Lightning attachment to aerospace materials is a two-way non-linear process in which the elastoplastic electromagnetic and thermal response of the material can alter the behaviour of the lightning arc. The Laboratory for Scientific Computing, in collaboration with Boeing, has worked on techniques for capturing this behaviour in a single model [1] with ionisation of the plasma dealt with through a 19-species equation of state [2]. These models focus on attachment to carbon composite materials, which, due to low electrical and thermal conductivity, can be more vulnerable to energetic input from a lightning strike. A number of projects are offered to further develop these models, for example considering an anisotropic model of the aerospace materials, utilising mesh refinement techniques to consider the long time scale thermal effects or through novel interface treatments allowing delamination to be studied.

[1]: S. Millmore and N. Nikiforakis. "Multi-physics simulations of lightning strike on elastoplastic substrates." Journal of Computational Physics 405 (2020): 109142.

[2]: F. Träuble, S. T. Millmore, and N. Nikiforakis. "An improved equation of state for air plasma simulations." Physics of Fluids 33.3 (2021): 036112.

**Dr Stephen Millmore, Laboratory for Scientific Computing, Department of Physics
Email: stm31@cam.ac.uk**

Whole-system simulations of the conditions within a Tokamak reactor offer a predictive tool for controlling operating conditions and understanding the interaction of the fusion plasma with the reactor walls. Additionally, the plasma can generate instabilities leading to energetically violent events and strong gradients. The Laboratory for Scientific Computing, in collaboration with Tokamak Energy, is working on numerical methods to allow for the simulation of the full range of behaviour within a Tokamak reactor and aims to couple this behaviour to the elastoplastic response of the reactor walls. A number of projects are offered to develop the underlying mathematical models and the corresponding algorithms on different aspects of the process ranging from simulating the formation of the plasma from the initial magnetic currents to predicting the response of resistive materials in direct contact with the ionised material.

**Dr Stephen Millmore, Laboratory for Scientific Computing, Department of Physics
Email: stm31@cam.ac.uk**

Gyrotrons producing millimetre waves offer a cost-effective technology for drilling through geological materials, in particular hot igneous rocks, for which the temperatures and material hardness are challenging for conventional drilling techniques. A key application of this drilling technology is deep geothermal energy. However, the lack of information from the bore hole during the drilling process means simulation data is essential for an accurate and efficient process. The aim of this project, which is in collaboration with Quaise, is to further develop the models suitable for the simulation of the large scale behaviour determining the effects of anisotropic geological configurations and interfaces between rock types [1], as well as local behaviour considering the flow and evaporation of rock, particulate transport away from the evaporation surface and plasma formation as a result of the millimetre wave source.

[1]: Z. Zhang, S. Millmore and N. Nikiforakis “Simulation of Millimetre Wave Evaporation of Rock” In preparation

**Dr Bartomeu Monserrat, Department of Materials Science & Metallurgy**

Email: bm418@cam.ac.uk

Email: bm418@cam.ac.uk

**Prof. Nikos Nikiforakis, Laboratory for Scientific Computing, Department of Physics
Email: nn10005@cam.ac.uk **

The aim of this project is the development and application of numerical algorithms and High Performance Computing methods for the simultaneous solution of the complex systems of nonlinear partial differential equations for the direct numerical simulation of four states of matter (Computational Multiphysics) at extreme states.

Examples include elastoplastic structural response due to detonation loading, lightning strike on aerospace composites, nonlinear events in nuclear fusion reactors and high-energy beam melting of geological materials.

More information about Prof. Nikiforakis's research can be found at: