There is no doubt that computational mechanics has been very successful in advancing the fields of engineering and science in general. Nevertheless, modern engineering problems often involve a complex interaction of physical phenomena and complicated, possibly evolving, geometries. Many times, physical situations are encountered for which traditional models are not appropriate. There is a need for advancement of all aspects of computational mechanics, from model development, to numerical methods, to details of implementation on emerging exascale computational facilities. I have experience and expertise in all of these steps.
As an assistant professor, my research plans will allow me to make valuable contributions to many areas relating to mechanical engineering and to provide relevant projects for graduate students in which the balance of abstraction and application can be chosen to suit each student's wishes and strengths. I have several projects in mind that are already underway or could be started immediately. I list some of them in the following sections.
Continuum and Rarefied Gas-Flow Prediction
My main areas of research to date are in the field of numerical prediction of continuum and non-equilibrium fluid-dynamic, heat-transfer, and thermodynamic phenomena. Most traditional engineering techniques have been developed for the prediction of thermodynamic and fluid-dynamic systems that only exist in local equilibrium. Applications for which traditional equilibrium models are not appropriate include high-atmosphere flight, space-propulsion systems, super- or hyper-sonic flows, space-weather prediction, and very small-scale flows, such as within micro-electromechanical systems.
The models I am developing offer an expanded region of physical validity as well as mathematical advantages as compared to the Navier-Stokes equations. In fact, even for many traditional applications, these new methods can be superior to the Navier-Stokes equations [1, 2, 3, 4, 5]. The particular techniques I use are moment closures of the Boltzmann equation. These moment closures lead to sets of stable first-order hyperbolic conservation-type partial differential equations (PDEs) for an expanded solution vector (beyond mass, momentum, and energy). This provides a natural treatment of thermal non-equilibrium effects such as anisotropic temperatures or pressures. The first-order hyperbolic balance-law form of the resulting systems yields immediate numerical advantages. For example, an extra order of spatial accuracy can be achieved for a given spatial stencil. Also, I have shown that moment closures offer increased insensitivity of solutions to grid irregularities . This is important for practical engineering applications with complex geometries for which the generation of very high-quality meshes can be tedious, when solution-directed adaptive mesh refinement is used and can significantly degrade grid quality, or when embedded moving-boundary treatments are used. Solutions for models that rely on second derivatives, such as the Navier-Stokes equations, are much more sensitive to grid smoothness.
The use of PDE-based approach also gives the possibility for PDE-constrained optimization, such as adjoint-based methods for non-equilibrium situations. These techniques are becoming an increasingly integral part of the engineering design process and are not compatible with stochastic particle-based techniques for rarefied-gas prediction, such as direct-simulation Monte Carlo (DSMC).
Through my research, I have demonstrated that techniques based on moment closures can provide very efficient flow software for the prediction of gas flows, both in and for significant departure from local thermodynamic equilibrium for many canonical flow situations [1, 2, 5, 6]. I now plan to apply my novel fluid models to the practical "real-world" engineering problems listed above and thoroughly demonstrate the various advantages that they bring as compared to traditional methods.
Radiation transport is a fundamentally important part of mechanical-engineering applications. It is an important component of most engineering problems involving heat-transfer and is the phenomena of interest for many situations. One area in which accurate radiation modelling and prediction is of paramount importance is nuclear-power generation. Radiative transport is also an area of interest for many medical applications. This includes medical imaging and radiative therapies for cancer treatment. All current methods for radiation-transport prediction (including discrete-ordinance methods, spherical harmonics, stochastic particle-based methods, and diffusion approximations) are inadequate or inefficient in some regard, even for some simple canonical problems . There is a need for improved methods.
There has been recent interest in the use of moment methods for radiative-transfer prediction, both in nuclear energy and medical fields [8, 9] as moment-based methods offer the promise of remedies for many of the issues suffered by traditional methods. The radiative-transfer and neutron-transport equations are very similar to the Boltzmann equation for gases and all of my expertise in kinetic theory and moment methods can be immediately applied. I am looking forward to extend my techniques into this field.
Multi-Phase Flows and Granular Gases
Kinetic theory can be used to model the behaviour of both multi-phase fluids, such as gases containing liquid droplets or particulate, and granular gases. Droplet dynamics are a difficult and important part of many engineering applications, such as internal-combustion-engines and propulsion-systems, as fuels are often injected as a liquid spray. Granular gases are important in many engineering fields, including the design of fluidized-bed burners and reactors. Particulate modelling is also very important for combustion systems, when soot formation and dynamics play an important role. Improved modelling of soot is necessary to reduce the emissions of any combustion system. Moment-closure approaches based on my investigations into hyperbolic moment equations can provide accurate models for these complicated situations. I am eager to extend my work into this field as, ultimately, more accurate simulations will lead to more efficient mechanical and energy-conversion systems.
The modelling of non-equilibrium plasmas involve the interaction between fluid flows and electro-magnetic effects. Effective modelling of plasmas is important in the study of astrophysical phenomena, electro-magnetic space propulsion systems, industrial situations involving electrical arcing, mass spectroscopy, plasma actuators for flow control, or nuclear-fusion power generation. Moment closures offer a natural way to couple fluid equations for ions and neutrals to electric fields which can be prescribed or can be determined through the solution of Maxwell's equations. Inter-particle effects such as ionization, dissociation, and other chemical processes can be readily handled through local source terms. I plan to demonstrate that moment-closure-based methods offer modelling advantages for plasmas when compared to equilibrium treatments while being more efficient than particle-based methods (as I have already shown for gases [1, 2].
Numerical Framework for Complex Systems
I am eager to construct a large-scale computational framework for the prediction of practical continuum and non-equilibrium engineering problems. The models I am developing are at a state that they are ready for application to large-scale engineering problems. Many practical applications involve the coupling of the various physical phenomena listed above (among others) and I have ideas for how a framework should be built such that smaller equation sets and solvers can be easily "plugged" together allowing simulation software for a particular complex system to be easily assembled without sacrificing computational efficiency. This framework will be useful for fundamental model development, investigations of special computational and numerical techniques for complex coupled physics, and studies of real-world engineering applications in which non-equilibrium fluid-dynamic and thermodynamic effects may be important.
The first-order balance-law form of my moment equations means the entire range of numerical methods for hyperbolic balance laws, such as Godunov-type finite-volume schemes and discontinuous-Galerkin schemes, can immediately be used. I have experience writing software that efficiently runs on large-scale distributed-memory computational facilities with tens of thousands of cores  and have made use of techniques ranging from solution-directed adaptive mesh refinement and embedded moving boundaries  to GPU acceleration. Future computational facilities will have a very wide array of hardware configurations, therefore, designing algorithms and developing codes that can efficiently make use of such heterogeneous configurations will become more and more important. I will develop a framework that can distribute work among whatever computational resources are available in order to optimize hardware usage.
Advanced Numerics for CFD
Although moment closures yield sets of equations in balance-law form, there remain issues to their numerical solution. Traditional numerical methods for conservation laws maintain positivity of certain entries in the solution vector, such as density and energy, however, physical realizability of moment states for non-equilibrium gases is more complex. For example, moment equations allow for anisotropic pressures through the inclusion of a pressure tensor in the solution vector. Physical realizability requires that this second-order tensor remains positive definite. This leads to a complicated restriction involving six of the conservation laws, and is difficult to satisfy in all computational applications , especially for highly non-equilibrium situations. I plan to develop modifications or extensions to traditional numerics for hyperbolic conservation laws that are specifically designed to allow for the reliable and robust solution of moment systems for highly non-equilibrium situations.
I am interested and experienced in every step of computational mechanics: modelling, numerics, and computational application. My research will make significant contributions to several fields within mechanical engineering and will provide appropriate and highly relevant projects for students and researchers that are of interest to both external funding agencies and private industry.
- McDonald, J. G. and Groth, C. P. T., "Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution", Continuum Mechanics and Thermodynamics, 2012, DOI:10.1007/s00161-012-0252-y.
- McDonald, J. G., Extended Fluid-Dynamic Modelling for Numerical Solution of Micro-Scale Flows, Ph.D. thesis, University of Toronto, 2011.
- Groth, C. P. T. and McDonald, J. G., "Towards physically-realizable and hyperbolic moment closures for kinetic theory", Continuum Mechanics and Thermodynamics, 21: 467-493, 2009.
- Suzuki, Y., Khieu, L., and van Leer, B., "CFD by first order PDEs", Continuum Mechanics and Thermodynamics, 21: 445-465, 2009.
- McDonald, J. G., Sachdev, J. S., and Groth, C. P. T., "Gaussian moment closure for the modelling of continuum and micron-scale flows with moving boundaries", in H. Deconinck and E. Dick (eds.) Proceedings of the Fourth International Conference on Computational Fluid Dynamics, ICCFD4, Ghent, Belgium, July 10-14, 2006, pages 783-788, Springer-Verlag, Heidelberg, 2009.
- McDonald, J. G. and Groth, C. P. T., "Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure", Paper 2008-0691, AIAA, January 2008.
- Brunner, T. A., "Forms of approximate radiation transport", Paper SAND2002-1778, Sandia National Laboratories, July 2002.
- Barnard, R., Frank, M., and Herty, M., "Optimal radiotherapy treatment planning using minimum entropy models", , 2012, arXiv:1105.5261v1.
- Hauck, C. D., "High-order entropy-based closures for linear transport in slab geometry", Communications in Mathematical Science, 9: 287-205, 2011.