Multiplication Matrices and Ideals of Projective Dimension Zero

We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.     

A simple method for crack growth in mixed mode with X-FEM

In this paper, a new method for level set update is proposed, in the context of crack propagation modeling with the extended finite element method (X-FEM) and level sets. Compared with the existing methods, such as the resolution of the Hamilton–Jacobi equations, this new method is much simpler because it does not required complex manipulations of the level sets. This method, called the “projection” method, uses both a classical discretization of the surface of the crack (segments for 2d cracks and triangles for 3d cracks) and a level set representation of the crack. This discretization is updated with respect to the position of the new crack front. Then the level sets are re-computed using the true distance to the new crack, by an orthogonal projection of each node of the structure onto the new crack surface. Then, numerical illustrations are given on 2d and 3d academic examples. Finally, three illustrations are given on 3d industrial applications.     

Probing the structure of complex solids using a distributed computing approach—Applications in zeolite science

We demonstrate the viability of distributed computing techniques employing idle desktop computers in investigating complex structural problems in solids. Through the use of a combined Monte Carlo and energy minimisation method, we show how a large parameter space can be effectively scanned. By controlling the generation and running of different configurations through a database engine, we are able to not only analyse the data “on the fly” but also direct the running of jobs and the algorithms for generating further structures.As an exemplar case, we probe the distribution of Al and extra-framework cations in the structure of the zeolite Mordenite. We compare our computed unit cells with experiment and find that whilst there is excellent correlation between computed and experimentally derived unit cell volumes, cation positioning and short-range Al ordering (i.e. near neighbour environment), there remains some discrepancy in the distribution of Al throughout the framework. We also show that stability-structure correlations only become apparent once a sufficiently large sample is used.