Numerical modelling and simulation of physical processes and phenomena has gained in popularity in recent years mainly due to the development of efficient numerical algorithms and the emergence of new parallel computing architectures. This led to a major shift in scientific treatment of complex systems and phenomena to an integrated approach where analytical, experimental and numerical modelling interact closely to achieve the common goal. Although the initial cost of developing effective tools for numerical modelling is potentially high, the results that can be produced by a numerical model can be used to focus the experimentation, leading to a considerable time and cost cutting.
We are interested in continuum modelling of multi-physics systems, based on the finite element method. Multi-physics systems are the systems with constitutive parts arising from different disciplines (for example, fluid-solid interaction, magnetohydrodynamics, nano-fluidics, thermodynamics). Although in many cases a well-developed algorithms for numerical solution of single-physics problems exist (for example the Navier-Stokes equations in fluid mechanics or Maxwell's equations in electromagnetics), the efficient solution of multi-physics problems still represents a considerbale challenge.
Our approach in modelling multi-physics systems is to perform a monolithic discretisation of the spatial part of the problem, and to solve the resulting system of differential algebraic equations by an implicit adaptive timestepping algorithm, based on the predictor-corrector method. The main difficulty that arises in this approach is to solve large, sparse, non-symmetric linear systems that are ill-conditioned. When modelling time-dependent problems, these systems need to be solved repeatedly large number (possibly thousands) of times. In such cases the use of iterative solution strategy is inevitable. In order to achieve a nearly-optimal scaling of Krylov solvers, one needs to develop an efficient preconditioning strategy based on multigrid methods. Our research is focused primarily on that problem.
Current research involves collaborative work with School of Mathematics. With Prof. Matthias Heil we work on the development and efficient parallel implementation of preconditioned iterative solvers for fluid-solid interaction problems. These problems arise in biomechanical engineering when modelling blood vessels and broncihials. The preconditioned iterative solvers are implemented using general parallel block preconditioning framework and are integrated in the OOMPHLIB - an object oriented multi-physics library.
With Prof. David Silvester and Prof. Howard Elman (University of Maryland) we work on efficient algorithms for the numerical simulation of thermally bouyed flows (fluid flows in the presence of the temperature gradient). The applications of this model are vast: heat exchange systems, environmental modelling (oceanic currents, continental drift, katabaric winds), semiconductor fabrication, superconductivity. We developed an efficient timestepping scheme and AMG-based block preconditioner based on the reuse of efficient preconditioners for the Navier-Stokes equations. This model is a part of the IFISS package.
Future plans include the development (within the OOMPHLIB framework) of the single-physics modules for electromagnetics, micromagnetics and nanofluidics and study of the multi-physics problems that include these models.