SUMMARYConformal refinement using a shrink and connect strategy, known as pillowing or buffer insertion, contracts and reconnects contiguous elements of an all-quadrilateral or an all-hexahedral mesh in order to locally increase vertex density without introducing hanging nodes or non-cubical elements. Using layers as shrink sets, the present method automates the anisotropic refinement of such meshes according to a prescribed size map expressed as a Riemannian metric field. An anisotropic smoother further enhances vertex clustering to capture the features of the metric. Both two-and three-dimensional test cases with analytic control metrics confirm the feasibility of the present approach and explore strategies to minimize the trade-off between element shape quality and size conformity. Additional examples using discrete metric maps illustrate possible practical applications. Although local vertex removal and reconnection capabilities have yet to be developed, the present refinement method is a step towards an automated tool for conformal adaptation of all-quadrilateral and all-hexahedral meshes.