2 and 3.3.3, respectively. Here, the particular focus is on metric formation. In general, more information can be found in Applied Modelling and Computation Group (2011) and the cited references. It is also noted that Fluidity-ICOM treats all input meshes in the same manner and uses an unstructured data structure to represent both structured and unstructured meshes, hence the key distinction is between fixed and adaptive meshes. In Fluidity-ICOM, a metric, represented by a symmetric positive definite tensor, is constructed (George and Borouchaki, 1998). This metric allows information C59 cost about the system

state to be contained in a form that can be used to guide the mesh optimisation step, Section 3.3.2. More specifically, given a metric, M  , the aim of the mesh optimisation step is to form a mesh, MM, with edges, vv, such that equation(5) ||v||M=vTMv=1,∀v∈M.That is to say, all edges in the mesh have unit length when measured with respect to the metric, M. The metric can be viewed as the continuous analogue of the mesh, describing

both the shape and size of the elements ( Loseille and Alauzet, 2011a). The choice of metric is, therefore, fundamental to the way in which the mesh adapts and where mesh resolution will be placed. Three metrics Trichostatin A order are considered here each of which is based on the Hessian of a solution field(s), H   (matrix of second-order derivatives), and a user-defined weight, ∊∊, that can vary spatially and/or temporally.

The form of each metric is motivated by interpolation error theory and they are chosen such that, for the exact Hessian, the metrics provide a bound for the interpolation error of the solution field under a selected norm. The first metric, M∞M∞, is given by equation(6) M∞(x)=|H(x)|∊(x),(e.g. Frey and Alauzet, 2005 and Pain et al., 2001), where |H(x)||H(x)| is a modified Hessian: equation(7) |H(x)|=Q(x)T|Λ(x)|Q(x),|Λ(x)|ij=|λi(x)|i=j0i≠jwith λiλi the eigenvalues of the Hessian and Q   the corresponding matrix of normalised eigenvectors. Information selleck about both the magnitude and direction of the curvature of the field is therefore included, via |Λ||Λ| and Q  , respectively, and facilitates the formation of anisotropic elements. If this metric is used and the adaptivity criteria, Eq. (5), is satisfied then, for an exact Hessian, a bound for the interpolation error is provided for a mesh element, ΩeΩe, under the L∞L∞ norm ( Frey and Alauzet, 2005). In practice, areas with a high curvature of a field (large second-order derivatives) and therefore larger eigenvalues, will demand refinement of the mesh, Eqs. (5), (6) and (7). Reducing the solution field weight will also promote more mesh refinement. Conversely, lower curvature and/or a larger solution field weight will demand coarsening of the mesh. The second metric, MRMR, has the form equation(8) MR(x)=1∊(x)|H(x)|max(|f(x)|,fmin)=M∞max(|f(x)|,fmin),(Castro-Díaz et al.