A variational grid optimization method based on a local cell quality metric
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Computational grid optimization, correction, improvement and remeshing techniques have become increasingly important as the application problem and domain complexity in creases. It is well recognized that distorted elements may degrade accuracy of ﬁnite element and ﬁnite volume simulations or cause them to fail. Hence, automatically generated grids containing millions of cells, created to ﬁt a domain with complex geometry and adapt to features of diﬀerent scales, often require correction before they can be eﬀectively used for a numerical simulation. In this work a new variational grid smoothing formulation is devel oped and an extensive study of its mathematical properties, applicability and limitations is performed. The approach is based on a local cell quality metric, which is introduced as a function of the Jacobian matrix of the fundamental map from the reference cell. The math ematical properties of the local quality measure are analyzed and new theoretical results are proved. The grid improvement strategy is formulated as an optimization problem and a modiﬁed Newton scheme is used in the optimization algorithm which is implemented in a new software package. The eﬀectiveness of the algorithm is tested on several representative v grids and for diﬀerent transport application problems. The resulting methodology is applicable to general unstructured hybrid meshes in 2 and 3 dimensions. It overcomes several diﬃculties encountered by other smoothing algo rithms, such as eﬀects of changing valence (number of cells sharing the same node). The formulation includes extensions to unfolding, adaptive redistribution, treatment of tangen tially “sliding” boundary nodes and hanging nodes, as well as elements with curved edges or surfaces, commonly used to provide better ﬁt of domain boundaries or interfaces. The above techniques are applied to a set of mathematically representative prob lems including problems of geometric design as well as transport processes with the aim of studying the eﬀect of the smoothing approach on the solvability and accuracy. Both 2D and 3D test problems are considered, including a moving mesh Lagrangian formulation for a ﬂuid interface problem, non-Newtonian blood ﬂow in curved branched pipes and a brain mapping/deformation problem. The associated numerical simulations are made on both serial and parallel PC cluster systems.