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A mesh refinement algorithm for arbitrary tetrahedral meshes has been developed. The algorithm is suitable for use in a variety of adaptive mesh refinement schemes and has the following features: (1) it can be applied to both optimal (Delaunay) and nonoptimal meshes, (2) new nodes are inserted using a perturbed edge bisection to prevent crossing edges, and (3) the Delaunay criterion is applied locally over each tetrahedron selected for refinement. The advantage of the local Delaunay subdivision is that it decouples the subdivision process, which reduces computation time. The method has been successfully applied to several magnetostatic problems modeled using first-order tetrahedra, and has produced refined meshes of over 215000 elements.