The refinement or coarsening of the mesh is still guided by the curvature

Alectinib price of the field. However, a scaling by the local magnitude of the field is now included in the metric. The final metric is obtained by consideration of the interpolation error in the LpLp norm, p∈[1,∞)p∈[1,∞). The general metric, denoted MpMp, has the form equation(9) Mp(x)=1∊(x)(det(|H(x)|))-12p+n|H(x)|=(det(|H(x)|))-12p+nM∞,(Chen et al., 2007 and Loseille and Alauzet, 2011b), where n   is the spatial dimension of the problem. Since det|H|=∏i|λi|det|H|=∏i|λi|, a scaling by a measure of the magnitude of the curvature of the field is included in the metric. The extent to which det|H|det|H| influences the metric is determined by the choice of p  . As p   is reduced, smaller scales are given more weight in the metric and as a result are better represented ( Loseille and Alauzet, 2011b). In the limit p→∞p→∞, M∞M∞ is recovered. The work of Loseille and Alauzet (2011b) shows that the influence of smaller scales in the metric rapidly decreases

as p   increases and their good results for p=2p=2 motivates the use of this value here. Hence, the third and final metric is given by equation(10) M2(x)=1∊(x)(det(|H(x)|))-16|H(x)|=(det(|H(x)|))-16M∞. In Fluidity-ICOM, the user chooses which solution fields a metric will be formed for and, therefore, which fields the mesh will adapt to. If the user chooses selleck inhibitor to adapt to multiple solution fields, a metric, MfMf, is formed for each chosen solution field, f  . The final metric, M  , is then obtained from a superposition of the metrics for individual fields M=⋂fMfM=⋂fMf ( Castro-Díaz et al., 1997). The user must also specify minimum and maximum edge lengths and this information is Ponatinib cost included through a restriction

on the eigenvalues of |H||H| (e.g. Pain et al., 2001). In addition, the user can provide an upper and/or lower bound on the number of mesh vertices. If the adaptive algorithm is configured appropriately, this bound should not be reached. Given a metric, the aim of the mesh optimisation step is to satisfy the criteria, Eq. (5) and thereby optimise the mesh for the current system state. The mesh is modified through a series of local topological and geometrical operations which, in two dimensions in Fluidity-ICOM, are performed using the algorithms of Vasilevskii and Lipnikov (1999). The operations include edge-collapsing, edge-splitting, edge-swapping and vertex-movement. More details and diagrams can be found in Pain et al., 2001, Piggott et al., 2009 and Vasilevskii and Lipnikov, 1999. Once the mesh optimisation stage has been performed, solution fields have to be interpolated between the pre- and post-adapt meshes. The interpolation methods available in Fluidity-ICOM fall into two categories. The first is referred to as consistent interpolation ( Applied Modelling and Computation Group, 2011).