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In simulation of crack propagation,constructing conforming mesh for the evolving crack domain,especially in 3D,still represents a difficult task.To address it,we introduce a novel method,called the successive node snapping(SNS)scheme,to construct conforming meshes for an evolving crack domain in 2D and 3D,with a given simplicial background mesh(triangular or quadrangular mesh in 2D,tetrahedral mesh in 3D).