Summary: A stress analysis using a mesh-free method on a cracked elastic medium needs a partition of unity for a non-convex domain whether it is defined explicitly or implicitly. Constructing such partition of unity is a nontrivial task when we choose to create a partition of unity explicitly. We further extend the idea of the almost everywhere partition of unity and apply it to linear elasticity problem. We use a special mapping to build a partition of unity on a non-convex domain. The partition of unity that we use has a unique feature: the mapped partition of unity has a curved shape in the physical coordinate system. This novel feature is especially useful when the enrichment function has polar form, \(f(r, \theta) = r^\lambda g(\theta)\), because we can partition the physical domain in radial and angular directions to perform a highly accurate numerical integration to deal with edge-cracked singularity. The numerical test shows that we obtain a highly accurate result without refining the background mesh.