A mesh metric field has been used to represent the element shape and size distribution. Definition, significance of the mesh metric field and the conformity criteria between the mesh and the mesh metric field are given. At low level, the mesh adaptation procedure operates as a sequence of mesh modification operators ordered to make the mesh conform to a given mesh metric field.

The mesh modification procedure consists of four related high level components: mesh refinement, mesh coarsening, projecting boundary vertices onto curved geometry and element shape correction. The mesh is efficiently aligned to the mesh metric field by incremental refinement and coarsening based on edge length analysis with respect to the mesh metric field. Several techniques have been developed to ensure effective alignment to the mesh metric field, for example, simultaneous split of a set of longest mesh edges, collapsing the shortest mesh edge every other vertex, diagonal edge selection in case of ambiguity and the use of lookup tables. The curved geometry approximation issue is addressed by the procedure of projecting boundary vertices onto geometry. The procedure first effectively projects as many vertices as possible using local mesh modifications. The possible remaining un-projected vertices after mesh modifications can be dealt with by a generalized local cavity re-meshing procedure. The element shape correction procedure improves the alignment to mesh metric field by eliminating sliver elements. Element shape analysis techniques guide the effective determination of the best mesh modification.

Two approaches of adaptively specifying mesh metric field are provided, one using second derivatives and the other using both second derivative and gradient information. Both have been integrated with the mesh adaptation procedure for anisotropic adaptive simulation on 3-D general geometries. Results of adaptivity show the effectiveness and the alignment of adapted meshes to both geometry and mesh metric field.