## QM/MM Multiscale Methods
Understanding locality of interaction, or other forms of sparsity, in *ab
initio* models can be effectively exploited to construct highly efficient and
accurate multi-scale schemes e.g. of QM/MM type. In QM/MM schemes we model
regions of interest, such as a dislocation core in the following figure, using
an electronic structure model, while the material bulk is modelled with a
computationally inexpensive interatomic potential model.
Upon increasing the QM core region, the accuracy of the simulation should, in
principle, increase, such as the following figure:
However, various artefacts in commonly used QM/MM schemes prevent this. The
main new idea that made this figure possible is that we construct interatomic
potentials (or forces) that are *tuned to interact well with the QM model*
rather than using "off-the-shelf potentials". This yields new QM/MM schemes with
rigorous convergence rates in terms of the QM region size. [\[56\]](index.php?page=publications#2015-qmtb1), [\[57\]](index.php?page=publications#2015-qmtb2),
[\[58\]](index.php?page=publications#2015-tfw)
## Higher-order Far-field Boundary Conditions
Most multi-scale embedding schemes (A/C, QM/MM, . . .) schemes can be understood
within a single framework: a coarse model for a far-field is used to provide a
boundary condition for a core region of interest. It turns out that, for
crystalline defects, boundary conditions of arbitrary accuracy can be
constructed *almost* analytically (at least in principle), without ever having
to resort to the complex mechanisms and many pitfalls of multi-scale schemes.
We achieve this by constructing an expansion of an atomistic model in
terms of higher-order continuum models, e.g., the first three terms in the
expansion are typically (1) linearised elasticity, (2) nonlinear elasticity,
(3) strain-gradient elasticity. This yields high accuracy simulation results
with small computational domains, e.g., in the following figure for a screw
dislocation (bcc, W, 111, EAM); see
[\[73\]](index.php?page=publications#2017-bcscrew),
[\[48\]](index.php?page=publications#2013-defects) for more details.
## 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. For representative
publications see [\[45\]](index.php?page=publications#2013-atc.acta),
[\[48\]](index.php?page=publications#2013-defects),
[\[53\]](index.php?page=publications#2014-bqce),
[\[54\]](index.php?page=publications#2014-bgfc).
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.
Two **open Problems** I am still interested in:
* A/C coupling and in general coarse-graining for transition rates, bifurcations, and other critical phenomena
* A/C coupling for cracks; there are some surprisingly interesting analytical questions in this

## Nonlinear Elasticity and the Lavrentiev Gap Phenomenon
In the past I have worked on the numerical approximation of nonlinear elasticity, understood as an energy minimisation problem,
$$
\bar{u} \in \arg\min\big\{ I[u] \big| u \in A \big\}, \qquad \text{where} \qquad I[u] = \int_\Omega W(x, u , \nabla u)\,dx
$$

and $A$ is some *admissible set*. A major difficulty in this area is the **Lavrentiev gap phenomenon** which is said to occur if
$$
\inf \big\{ I[u] \big| A \cap W^{1,\infty} \big\} \gneq \inf \big\{ I[u] \big| u \in A \big\}.
$$

That is, 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 [\[18\]](index.php?page=publications#2011-SINUM_ANCFEM), [\[20\]](index.php?page=publications#2011-IMAJNA-lavrentiev_nc) 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).
Although I have not worked on this topic for some time, there is a particularly difficult **open problem** which I would love to solve, or see solved:
* Construct a numerical method that is capable of approximating global minimizers for any well-posed variational problem (e.g., fitting within the [Ball, 1977] theory) with polyconvex stored energy function.
## Miscellaneous
I also work, or have worked, on the following topics:
* Adaptive finite element methods for phase field models of fracture
* Phase field models for dynamic crack propagation
* Griffith model for fracture
* Discontinuous Galerkin finite element methods
* Gradient flows, in particular nonlinear viscoelasticity
* Large-scale optimisation
* Convergence of adaptive finite element methods
* Numerical enclosure methods for elliptic [\[5\]](index.php?page=publications#2009-SINUM-ApostEx) and parabolic problems
* Optimal control