Atomistic/Continuum Multiscale Methods
My main research interest is coarse graining of atomistic models for solids, in particular the construction and analysis of continuum approximations and of atomistic-to-continuum coupling methods (a/c methods, quasicontinuum methods). The idea of a/c methods is to use computationally expensive atomistic models to describe only those regions of a computational domain that require atomistic accuracy, e.g., the neighbourhood of a crystal defect, and to use a coarse-grained continuum model to describe the elastic fields. At least in principle, this process can yield models with near atomistic accuracy at a significantly reduced computational cost. My focus so far has been on the theoretical foundations of these coarse graining techniques.
Together with my collaborators I have mainly worked on the error analysis of different variants of a/c methods. For example, the above figures show the computation of a micro-crack in a 2D model problem. The error graphs (which are consistent with our theory) show how different a/c methods lead to different convergence rates (including O(1)). This is one of the simplest crystal defects, and much more needs to be done to verify the effectiveness of a/c methods.
Open Problems I am interested in at the moment:
Nonlinear Elasticity and the Lavrentiev Gap PhenomenonIn the past I have worked on the numerical approximation of nonlinear elasticity, understood as an energy minimisation problem. A major difficulty in this area is an approximation problem related to the Lavrentiev gap phenomenon: for certain singular variational problems the infimum of the energy taken over Lipschitz functions can be strictly larger than the infimum taken over the entire admissible class. This means, in particular, that conforming finite element methods are incapable of detecting the global minimizers. I have shown that at least for convex problems (unfortunately this excludes elasticity) one can overcome this by using the non-conforming Crouzeix-Raviart finite element space.
The following figure shows a problem proposed by Foss, Hrusa and Mizel, solved by (mesh-adaptive versions of) the conforming P1 finite element method and the non-conforming Crouzeix-Raviart finite element methods. The left graph clearly shows a gap between the minimal energy (plotted against number of degrees of freedom in the FEM mesh).
I am still interested in solving the following Open Problem:
MiscellaneousI also work, or have worked, on the following topics: