Enriched finite element methods such as the Generalized Finite Element Method (GFEM), the eXtended Finite Element Method (XFEM), the cut-cell finite element method, the Interface-enriched Generalized Finite Element Method (IGFEM), and the Conforming Decomposition Finite Element Methods (CDFEM) are powerful tools for coupled multiphase and multimaterial problems with moving interfaces. In CDFEM, level sets are used to describe the domain of each material or phase. Nodes are added at the intersection of each level set surface with the edges of the input mesh, and a conforming mesh is generated automatically. This allows the weak and strong discontinuities across the interfaces to be captured using standard finite element methods. In recent work, a new strategy has been developed for automated interface conforming tetrahedral mesh generation, which produces higher quality meshes than standard CDFEM techniques. The method is termed the Conforming Transient h-r Unstructured Adaptive Mesh Refinement (cThruAMR). cThruAMR uses a combination of h-adaptivity and r-adaptivity to generate high quality meshes that conform to a moving interface. The term h-adaptivity is used for refining or cutting the mesh. Conforming h-adaptivity is used in CDFEM to capture dynamic topology problems. The term r-adaptivity is used for moving the nodes of the mesh to capture a desired feature. By combining h and r adaptivity, cThruAMR can produce high quality meshes even for dynamic topology problems. A new low diffusion cThruAMR method has been developed that has lower error than the initial implementation of the method. This focus of this talk is the description of cThruAMR for capillary hydrodynamics including the strategies employed for r-adaptivity, level set advection, dynamic DOF handling, and interfacial boundary conditions. The method is verified using multiple benchmark problems and the method is shown to produce optimal rates of convergence. *Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.