The　numerical　manifold　method　falls　into　the　category　of　the　partition　of　unity　methods.　In　order　to　enhance　accuracy,　high　order　polynomials　can　be　specified　as　the　local　approximations.　This,　however,　would　incur　rank　deficiency　of　the　stiffness　matrix.　In　this　study,　a　local　displacement　approximation　is　constructed　over　a　physical　patch　generated　from　a　four　quadrilateral　mathematical　mesh.　All　the　degrees　of　freedom　are　physically　meaningful.　The　stresses　are　continuous　at　all　nodes,　suggesting　that　no　stress　polish　is　required.　The　proposed　approximations　have　the　same　accuracy　as　the　first-order　polynomials,　but　no　linear　dependency　inherent　in　the　latter.　(C)　2016　Elsevier　Ltd.　All　rights　reserved.