A simple Hessian-based 3D mesh adaptation technique with applications to the multigroup diffusion equations Jean C. Ragusa * Texas A&M University, Department of Nuclear Engineering, College Station, TX 77843-3133, USA article info Article history: Received 20 February 2008 Accepted 23 June 2008 Available online 12 August 2008 abstract A simple error estimation and mesh adaptation procedure is proposed for the multigroup multi-dimen- sional diffusion equations. The procedure estimates the values of the second-order derivatives of the cur- rent numerical solution to drive the mesh reﬁnement. Different spatial meshes are obtained for each component of the multigroup ﬂux in order to follow the physics appropriately. This requires the proper handling of the group coupling terms arising in the multigroup diffusion equations. Results are presented in 2D and 3D. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction In the last decade or so, mesh adaptation techniques have been widely developed and implemented in many engineering disci- plines. One of the key advantages of mesh adaptation is that it al- lows for numerical solution to be automatically computed on a sequence of successively adapted (reﬁned) meshes, guaranteeing a ﬁnal solution that is typically tightly converged with fewer un- knowns and shorter CPU times than numerical solutions obtained with uniform mesh reﬁnement techniques (Demkowicz, 2007; So- lin, 2005). Yet, in the realm of reactor analysis, only limited efforts have been carried out to implement mesh adaptivity for neutronics applications. For instance, Zhang and Lewis (2001, 2002), applied a mesh adaptation strategy, known as p-reﬁnement where the poly- nomial order is selectively increased, but their published work only dealt with one-energy-group approximations. Ragusa et al. investi- gated an hp-adaptive strategy, where both a local mesh subdivision and a polynomial order increase are allowed, for the multigroup diffusion equations in 1D (Wang and Ragusa, 2006) and in 2D (Wang and Ragusa, accepted for publication). They found that their implementation of hp-adaptivity was, to some extent, sub-optimal, mostly due to the lack of simple error estimators for hp-strategies. Indeed, they used the difference between a ﬁne mesh solution, U ﬁne , and its projection onto the coarser current adapted mesh, P coarse U ﬁne , to drive the error estimation. Hence, this required the computation of the more expensive solution, U ﬁne , at every iteration of the mesh adaptation process in order to provide only an estimate of the coarse mesh error err coarse (this is easily demon- strated using P coarse U ﬁne  U ﬁne  P coarse U ﬁne  U true  (U ﬁne  U true )= err coarse  err ﬁne  err coarse ). Even though their hp-results showed signiﬁcant savings in terms of memory and some apprecia- ble gains in terms of CPU, it is somehow unsatisfying to use a ﬁner mesh solution only to drive the reﬁnement on the coarser mesh. The work presented here deals with this aspect; we developed and used an error estimator for the current adapted mesh based on the numerical solution on that mesh. This estimator is based on the estimation of the second derivatives of the numerical solu- tion. Such an estimator is simple to compute and implement. We limit our studies to linear ﬁnite element solutions applied to Carte- sian 2D and 3D meshes with local adaptive reﬁnements that lead to the subdivision of quadrilateral elements into four smaller ele- ments (2D case) and hexahedral elements into eight smaller ele- ments (3D case). Hence, the sequence of adapted meshes are irregular meshes (that is, meshes with hanging nodes) that belong to a hierarchy of adapted meshes, with the lowest level in the hier- archy being the initial (unadapted) structured Cartesian grid we started with. Additionally, as in our previous works, we speciﬁcally deal with multigroup approximations (as opposed to only one-group approx- imation) because we believe mesh adaptivity should be viewed and analyzed in that context for the following obvious reasons: (1) Each component of the multigroup scalar ﬂux can greatly vary in smoothness because of the disparity in cross sections between groups. Typically, fast-group ﬂuxes are much smoother than thermal-group ﬂuxes. Hence, using a single spatial mesh for all energy groups is far from optimal; indeed, it is easily understood that such an mesh would be too ﬁne for some groups and, hence, would not be a sensible usage of the available resources. As we have demonstrated in Wang and Ragusa (2006, accepted for publication), mesh adaptivity in the context of multigroup equations should be performed using group-dependent meshes, where each component of the multigroup ﬂux is computed using a mesh tailored for that speciﬁc group. 0306-4549/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2008.06.008 * Tel.: +1 979 862 2033; fax: +1 979 845 6443. E-mail address: ragusa@ne.tamu.edu Annals of Nuclear Energy 35 (2008) 2006–2018 Contents lists available at ScienceDirect Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene